2021

[1,2,3]Niccolò Bucciantini

1]INAF Osservatorio Astrofisico di Arcetri, Largo Enrico Fermi 5, 50125
Firenze, Italy 2]Dipartimento di Fisica e Astronomia, Università degli Studi
di Firenze, Via Sansone 1, 50019 Sesto Fiorentino (FI), Italy 3]Istituto
Nazionale di Fisica Nucleare, Sezione di Firenze, Via Sansone 1, 50019 Sesto
Fiorentino (FI), Italy 4]INAF Istituto di Astrofisica e Planetologia Spaziali,
Via del Fosso del Cavaliere 100, 00133 Roma, Italy 5]INAF Osservatorio
Astronomico di Cagliari,Via della Scienza 5, 09047 Selargius (CA), Italy
6]Department of Physics and Kavli Institute for Particle Astrophysics and
Cosmology, Stanford University, Stanford, California 94305, USA. 7]Istituto
Nazionale di Fisica Nucleare, Sezione di Pisa,Largo B. Pontecorvo 3, 56127
Pisa, Italy 8]RIKEN Cluster for Pioneering Research, 2-1 Hirosawa, Wako,
Saitama 351-0198, Japan 9]Hiroshima Astrophysical Science Center, Hiroshima
University,1-3-1 Kagamiyama, Higashi-Hiroshima, Hiroshima 739-8526, Japan
10]Yamagata University, 1-4-12 Kojirakawa-machi, Yamagata-shi 990-8560, Japan
11]Dipartimento di Fisica, Università di Pisa; Largo B. Pontecorvo 3, 56127
Pisa, Italy 12]Center for Astrophysics, Harvard Smithsonian; 60 Garden St,
Cambridge, MA 02138, USA 13]NASA Marshall Space Flight Center; Huntsville, AL
35812, USA 14]MIT Kavli Institute for Astrophysics and Space Research,
Massachusetts Institute of Technology; 77 Massachusetts Avenue, Cambridge, MA
02139, USA 15]Center for Research and Exploration in Space Science and
Technology, NASA/GSFC; Greenbelt, MD 20771, USA 16]NASA Goddard Space Flight
Center; Greenbelt, MD 20771, USA 17]Department of Astronomy, University of
Maryland; College Park, Maryland 20742, USA. 18]Deutsches Elektronen-
Synchrotron (DESY), 15738 Zeuthen, Germany 19]Guangxi Key Laboratory for
Relativistic Astrophysics, School of Physical Science and Technology; Guangxi
University, Nanning 530004, China 20]University of British Columbia;
Vancouver, BC V6T 1Z4, Canada 21]Instituto de Astrofísicade Andalucía, IAA-
CSIC; Glorieta de la Astronomía s/n, 18008 Granada, Spain 22]INAF Osservatorio
Astronomico di Roma, Via Frascati 33, 00078 Monte Porzio Catone, Roma, Italy
23]Space Science Data Center, Agenzia Spaziale Italiana,Via del Politecnico
snc, 00133 Roma, Italy 24]Dipartimento di Matematica e Fisica, Università
degli Studi Roma Tre; Via della Vasca Navale 84, 00146 Roma, Italy 25]Istituto
Nazionale di Fisica Nucleare, Sezione di Torino; Via Pietro Giuria 1, 10125
Torino, Italy 26]Dipartimento di Fisica, Università degli Studi di Torino; Via
Pietro Giuria 1, 10125 Torino, Italy 27]ASI - Agenzia Spaziale Italiana; Via
del Politecnico snc 00133 Roma, Italy 28]Istituto Nazionale di Fisica
Nucleare, Sezione di Roma “Tor Vergata”; Via della Ricerca Scientifica 1,
00133 Roma, Italy 29]Institut für Astronomie und Astrophysik, Universität
Tübingen; Sand 1, 72076 Tübingen, Germany 30]Space Research Institute of the
Russian Academy of Sciences; Profsoyuznaya Str. 84/32, Moscow 117997, Russia
31]Astronomical Institute of the Czech Academy of Sciences; Bočnı II 1401/1,
14100 Praha 4, Czech Republic 32]California Institute of Technology; Pasadena,
CA 91125, USA 33]Osaka University; 1-1 Yamadaoka, Suita, Osaka 565-0871, Japan
34]Department of Physics, Faculty of Science and Engineering, Chuo University;
1-13-27 Kasuga, Bunkyo-ku, Tokyo 112-8551, Japan 35]Institute for
Astrophysical Research, Boston University; 725 Commonwealth Avenue, Boston, MA
02215, USA 36]Department of Astrophysics, St. Petersburg State University;
Universitetsky pr. 28, Petrodvoretz, 198504 St. Petersburg, Russia 37]Physics
Department and McDonnell Center for the Space Sciences; Washington University
in St. Louis, St. Louis, MO 63130, USA 38]Finnish Centre for Astronomy with
ESO; 20014 University of Turku, Finland 39]Université de Strasbourg, CNRS,
Observatoire Astronomique de Strasbourg; UMR 7550, 67000 Strasbourg, France
40]Graduate School of Science, Division of Particle and Astrophysical Science;
Nagoya University, Furo-cho, Chikusa-ku, Nagoya, Aichi 464-8602, Japan
41]Department of Physics, The University of Hong Kong; Pokfulam, Hong Kong
42]Department of Astronomy and Astrophysics, Pennsylvania State University;
University Park, PA 16802, USA 43]Université Grenoble Alpes, CNRS, IPAG; 38000
Grenoble, France 44]Department of Physics and Astronomy, 20014 University of
Turku; Finland 45]INAF Osservatorio Astronomico di Brera; Via E. Bianchi 46,
23807 Merate (LC), Italy 46]Dipartimento di Fisica e Astronomia, Università
degli Studi di Padova; Via Marzolo 8,35131 Padova, Italy 47]Dipartimento di
Fisica, Università degli Studi di Roma “Tor Vergata”; Via della Ricerca
Scientifica 1, 00133 Roma, Italy 48]Mullard Space Science Laboratory,
University College London; Holmbury St Mary, Dorking, Surrey RH5 6NT, UK
49]Anton Pannekoek Institute for Astronomy & GRAPPA, University of Amsterdam;
Science Park 904, 1098 XH Amsterdam, The Netherlands

# Simultaneous space and phase resolved X-ray polarimetry of the Crab Pulsar
and Nebula

<EMAIL_ADDRESS>Riccardo Ferrazzoli  Matteo Bachetti  John Rankin
Niccolò Di Lalla  Carmelo Sgrò  Nicola Omodei  Takao Kitaguchi  Tsunefumi
Mizuno  Shuichi Gunji  Eri Watanabe  Luca Baldini  Patrick Slane  Martin C.
Weisskopf  Roger W. Romani  Andrea Possenti  Herman L. Marshall  Stefano
Silvestri  Luigi Pacciani  Michela Negro  Fabio Muleri  Emma de Oña Wilhelmi
Fei Xie  Jeremy Heyl  Melissa Pesce-Rollins  Josephine Wong  Maura Pilia  Iván
Agudo  Lucio A. Antonelli  Wayne H. Baumgartner  Ronaldo Bellazzini  Stefano
Bianchi  Stephen D. Bongiorno  Raffaella Bonino  Alessandro Brez  Fiamma
Capitanio  Simone Castellano  Elisabetta Cavazzuti  Stefano Ciprini  Enrico
Costa  Alessandra De Rosa  Ettore Del Monte  Laura Di Gesu  Alessandro Di
Marco  Immacolata Donnarumma  Victor Doroshenko  Michal Dovčiak  Steven R.
Ehlert  Teruaki Enoto  Yuri Evangelista  Sergio Fabiani  Javier A. Garcia
Kiyoshi Hayashida  Wataru Iwakiri  Svetlana G. Jorstad  Vladimir Karas
Jeffery J. Kolodziejczak  Henric Krawczynski  Fabio La Monaca  Luca Latronico
Ioannis Liodakis  Simone Maldera  Alberto Manfreda  Frédéric Marin  Andrea
Marinucci  Alan P. Marscher  Francesco Massaro  Giorgio Matt  Ikuyuki
Mitsuishi  C.-Y. Ng  Stephen L. O’Dell  Chiara Oppedisano  Alessandro Papitto
George G. Pavlov  Abel L. Peirson  Matteo Perri  Pierre-Olivier Petrucci  Juri
Poutanen  Simonetta Puccetti  Brian D. Ramsey  Ajay Ratheesh  Paolo Soffitta
Gloria Spandre  Toru Tamagawa  Fabrizio Tavecchio  Roberto Taverna  Yuzuru
Tawara  Allyn F. Tennant  Nicolas E. Thomas  Francesco Tombesi  Alessio Trois
Sergey Tsygankov  Roberto Turolla  Jacco Vink  Kinwah Wu  Silvia Zane  [  [  [
[  [  [  [  [  [  [  [  [  [  [  [  [  [  [  [  [  [  [  [  [  [  [  [  [  [
[  [  [  [  [  [  [  [  [  [  [  [  [  [  [  [  [  [  [  [

###### Abstract

The Crab pulsar and its nebula are among the most studied astrophysical
systems, and constitute one of the most promising environments where high
energy processes and particle acceleration can be investigated. They are the
only objects for which previous X-ray polarisation has been reported. We
present here the first Imaging X-ray Polarimetry Explorer (IXPE) observation
of the Crab pulsar and nebula. The total pulsar pulsed emission in the [2–8]
keV energy range is unpolarised. Significant polarisation up to 15% is
detected only in the core of the main peak. The nebula has a total space
integrated polarised degree of 20% and polarisation angle of 145∘. The
polarised maps show a large variation in the local polarisation, and regions
with polarised degree up to 45-50%. The polarisation pattern suggests a
predominantly toroidal magnetic field.

###### keywords:

X-rays: general, polarization, supernova remnants, pulsars: individual (Crab)

##  1 Introduction

The Crab pulsar (PSR B0531+21, PSR J0534+2200) and nebula (G184.6-5.8), born
out of the supernova SN1054, form one of the most interesting high energy
astrophysical systems, and one of the foremost environments where the physics
of compact objects, particle acceleration, and relativistic outflows can be
investigated Rees_Gunn74a ; Gaensler_Slane06a ; Hester08a ;
Buhler_Blandford14a .

The Crab pulsar (PSR) and nebula (PWN) are the only astrophysical system for
which integrated and/or phase-resolved X-ray polarisation has been reported by
various instruments Weisskopf_Silver+78a ; Forot_Laurent+08a ;
Moran_Shearer+13b ; Chauvin_Roques+13a ; Chauvin_Floren+17a ;
Vadawale_Chattopadhyay+18a ; Feng_Li+20a ; Long_Feng+21a ; Li_Produit+22a ,
suggesting a polarised degree (PD) in the pulses $\sim$20%, lower than in the
interpulse. The integrated X-ray PD in the off pulse (OP) due to the PWN is
consistently found to be $\sim$20% with typical polarisation angle (PA)
$\sim$130∘-140∘. The PWN polarimetry by OSO-8 has PD = 19.19% $\pm$ 0.97% and
19.50% $\pm$ 2.77% at 2.6 keV and 5.5 keV respectively, while PA = 156.36∘
$\pm$ 1.44∘ and 152.99∘ $\pm$ 4.04∘, at the same energies Weisskopf_Silver+78a
(all errors hereafter are at 1$\sigma$ confidence level, see also the
Methods).

The Crab PSR has a rotation period P = 33.7 ms, and spins down at a rate
$\dot{P}=4.21\times 10^{-13}$, corresponding to an effective surface dipole
magnetic field of $3.8\times 10^{12}$ G, and a spin-down luminosity $\dot{E}$
= $4.3\times 10^{38}$ erg s-1 Hester08a ; Buhler_Blandford14a . The estimated
distance is 2 kpc Trimble73a . Pulsed emission has been observed at all
wavelengths from radio up to TeV. The pulse shape is characterized by a main
pulse (P1), and an interpulse or second pulse (P2), whose positions in phase
show only a slight variation with energy. At optical frequencies and above, a
bridge (B) of emission is observed between them. The total unabsorbed pulsed
X-ray luminosity in the [2-10] keV band is $\sim 2.7\times 10^{-9}$ erg cm-2
s-1 Kaaret_Marshall+01a , while the photon index in the energy band [1-100]
keV is found to be in the range 1.4-2.2 Massaro_Campana+06a ;
Weisskopf_Tennant+11a ; Ge_Lu+12a ; Vivekanand21a . The photon index is phase
dependent: the emission is harder for B than for the peaks, and slightly
harder for P2 than P1. Optical and radio polarisation have been measured since
the PSR discovery. The OP emission (in optical in the phase range 0.78-0.84)
has a PD = 33%, and PA = 130∘ Sowikowska_Kanbach+09a . After OP subtraction
the average PD of the pulses is found to be 5.5% and the average PA is 96∘
Sowikowska_Kanbach+09a .

The Crab PWN has an apparent size of $6^{\prime}\times 4^{\prime}$ in the
optical band, corresponding to about $11\times 7$ ly at its estimated
distance. It has a broad band non-thermal spectrum, extending from radio to
TeV energies, due to synchrotron from high energy electrons and positrons up
to $\sim$150 MeV, and Inverse Compton above. The [2-8] keV integrated
luminosity is $\sim 1.8\times 10^{-8}$ erg cm-2 s-1 Kargaltsev_Pavlov08a  with
a photon index $\sim 2.12-2.15$ Mori_Burrows+04a . X-ray imaging shows a well
developed axisymmetric structure known as jet-torus Hester_Scowen+95a ;
Weisskopf_Hester+00a . Radio polarisation shows typical average values of the
PD in the range 5-10%, and PA = 150∘, with no correlation with the X-ray
features Aumont_Conversi+10a . High resolution optical polarimetry of the
inner region shows a PD = 40% and PA = 127∘ Moran_Shearer+13a .

##  2 Results

The Imaging X-ray Polarimetry Explorer (IXPE), the first mission devoted to
spatially-resolved polarisation measurements in the X-rays Weisskopf22d ;
Weisskopf_Soffitta+22x , was successfully launched on 2021 December 9. IXPE
observed the Crab PWN & PSR complex two times between February 21st and March
7th 2022 for a total on-source time of $\sim$92ks. Data were extracted and
analysed according to standard procedures: HEASOFT
6.30.1111https://heasarc.gsfc.nasa.gov/docs/software/heasoft/ was used to
perform barycenter correction, with the BARYCORR FTOOL, using the most recent
optical coordinates from the Gaia Data release
3222https://www.cosmos.esa.int/web/gaia/dr3, the DE421 JPL ephemeris and the
ICRS reference frame (see Table S1 in the Supplementary Material). ixpeobssim
V26.3.2 was used to do energy calibration, detector WCS correction, aspect-
solution corrections, and all further (unweighted) analysis Pesce-
Rollins_Lalla+19e ; Baldini_Bucciantini+22c , including phase folding at the
derived ephemeris. We also performed a coeval observation of the Crab PWN with
the CHANDRA Satellite (ObsId 23539, see the Methods for further details).

In Figure 1 (see also Table 1) we show the polarised properties of the PWN
derived by integrating all emission in a region within 2.5’ from the PSR.
Background contamination within this region is negligible. There is a
significant change in the PA between the low [2-4] keV and high [4-8] keV
energy band (true also for the OP emission - see Table S2 of the Supplementary
Material for a definition of the OP phase range). The change in PD is less
significant, with the higher energy band being slightly more polarised. The OP
phase emission is marginally more polarised than the total PWN & PSR emission,
as already suggested by a similar analysis of the OSO-8 data
Weisskopf_Silver+78a , while there is no evidence for even a marginal change
in PA, suggesting that the PSR has a net low level of polarisation, acting
mostly to reduce the total level of polarisation. The contribution of the PSR
unpulsed (DC) emission Tennant_Becker+01a  to the total OP emission, is
estimated to be less than 1%, and we can safely assume that the OP is mostly
of nebular origin. The OP polarisation angle is $\sim$145∘ , larger by
$\sim$20∘ with respect to the values reported in the literature for the PWN
symmetry axis, derived from fitting the X-ray jet-torus intensity maps
Weisskopf22d ; Ng_Romani04a . It is also smaller by $\sim$10∘ than the
previous OSO-8 measurements at a statistically significant level (see the
Supplementary Material Figure S1). Such discrepancies might simply reflect the
variability of the PWN, where structures are known to change in shape and
location over a typical timescale of a few years Schweizer_Bucciantini+13a .

In Figure 2 we show an intensity map of the Crab PWN from the coeval CHANDRA
ObsId 23539, the IXPE count map for the PWN$+$PSR complex in the [2-8] keV
energy range, and the IXPE count map in the same energy range, but computed
just for the OP emission.

For the phase-resolved analysis of the PSR we take events in the range [2-8]
keV and within 20” from the PSR itself to limit the PWN contamination. The
Stokes parameters of the OP emission have been subtracted (see the Methods for
the exact definition of the off pulse in terms of pulse phase, as well as the
other phase bins). In Figure 3 we plot the OP-subtracted light curve, in 200
equally spaced phase bins. For the polarisation analysis of the PSR emission
we opted for a variable phase binning, focusing on the peaks and bridge, in
order to get a finer sampling near P1 and P2. Figure 3 shows PSR normalised
Stokes parameters U/I and Q/I, for the phase bins of interest (see the
Supplementary Material Table S2 for further details). The OP emission in the
pulsar aperture has Q/I = -0.0106 $\pm$ 0.008, U/I = -0.241 $\pm$ 0.008,
corresponding to PD = 24.1% $\pm$ 0.8% and PA = 133.6∘ $\pm$ 1.0∘ . This is
significantly more polarised than the OP emission for the entire PWN, and the
PA is $\sim$10∘ smaller, indicative of a spatial variation of the polarisation
properties. The only phase bin showing a polarisation above the 3$\sigma$
confidence level is the center of P1 in the phase range [0.12,0.14], where the
OP subtracted emission has Q/I = -0.132 $\pm$ 0.025, U/I = -0.079 $\pm$ 0.025,
corresponding to PD = 15.4% $\pm$ 2.5% and PA = 105∘ $\pm$ 18∘ . There is no
significant change of the polarisation properties of this phase bin with
energy. Rapid PA variation might suppress the polarisation in these bins. The
total PSR normalised Stokes parameters are Q/I = -0.018 $\pm$ 0.019 and U/I =
-0.019 $\pm$ 0.019, confirming that the integrated PSR contribution serves
only to to reduce the polarisation of the entire complex.

In Figure 4 we show the total PWN & PSR map of PD in the [2-8] keV energy
range, obtained by smoothing the maps of Stokes parameters with a Gaussian
kernel, and cut at the 5$\sigma$ significance level (see the Supplementary
Material Figure S2), together with an intensity map where we have overlaid the
polarised magnetic field direction. The overall polarisation pattern agrees
with the general expectation for PWNe, where the synchrotron emission takes
place in the (mostly) toroidal magnetic field, originating from the pulsar
wind and compressed in the nebula, which sets the symmetry axis of the jet-
torus structure. The observed polarisation and emission patterns arise from
the interplay of the magnetic field geometry and bulk motion of the
relativistic plasma within the nebula itself, depending also on the
inclination of the nebular axis with respect to the line of sight. It is
indeed the presence of bulk motions directed toward and away from the observer
that creates the various bright arc-like features and makes the front side of
the torus brighter than the back. The results shown in Figure 4 agree with
this picture, assuming a symmetry axis inclined in the plane of the sky as
derived from X-rays Ng_Romani04a . The direction of the inferred magnetic
field broadly follows the shape of the emission torus (which extends also on
the back but is fainter due to Doppler de-boosting). There are two unpolarised
regions at the NE and SW edges of the main torus, where the polarised
direction varies rapidly within the point spread function. The overall PD map
shows a far stronger level of asymmetry with respect to the PWN axis than the
total intensity map, indicating possibly large variations in the amount of
magnetic turbulence within the PWN, or major distortions of the magnetic field
structure in the fainter outer regions. The more polarised regions are not
found in the center of the PWN, where there is a marginal contribution from
the PSR that lowers the PD, but north and south of the main torus, in regions
that do not correspond to any bright feature in X-ray. Based on smoothed maps
the peak PD in the Northern region is found to be 46%, with a PA of 163∘ , at
25$\sigma$ significance, while the peak in the southern region has PD of 51%,
with a PA 156∘ at 20$\sigma$ significance (see Methods Figure S3). Considering
instead two circular regions of 15” radius centered on the polarisation
maxima, we found for the northern region integrated quantities: Q/I = 0.37
$\pm$ 0.01 and U/I = -0.25 $\pm$ 0.01, corresponding to PD 45% $\pm$ 1% and PA
163.3∘ $\pm$ 0.8∘ , at 35$\sigma$ significance with an MDP at 99 confidence
level of 0.04, while the Southern region has Q/I = 0.30 $\pm$ 0.02 , U/I =
-0.37 $\pm$ 0.02, corresponding to PD 47% $\pm$ 2% and PA = 154∘ $\pm$ 1∘ , at
27$\sigma$ significance with an Minimum Detectable Polarisation (MDP) at 99%
confidence level of 0.05. These regions are far enough from the PSR that its
depolarising contribution is negligible. We caution the reader that due to
error in the reconstruction of the photons absorption point in the Gas Pixel
Detector (GPD), polarisation leakage can contaminate the Stokes maps, leading
to spurious polarisation patterns (this has no effect on integrated or OP
subtracted values). Preliminary estimates based on Montecarlo simulation of
the GDP response, indicate that this effect can at most be as high as 10% in
the outer regions of the PWN, and does not alter significantly our overall
findings.

##  3 Discussion

We report here the first phase- and space-resolved soft X-ray polarised
observation of the Crab PSR and PWN. Our results show that the total
polarisation of the pulsed signal is negligible. The consistency with optical
polarisation measure is marginal. Only the core of the main peak was found to
be significantly polarised. Marginal evidence for polarisation, below
3$\sigma$, in other phase bins are also reported. The low average polarization
is in contrast with the vast majority of the existing PSR models
Dyks_Harding04a ; Petri_Kirk05a  which typically predict polarisation fraction
in the pulsed emission of 40-80%. The model polarisation is generally
especially high in the B region. The peaks which are believed to be caustics
are typically de-polarised via rapid PA sweeps. In contrast we find our
highest PD in the core of P1. Moreover a simple PA swing does not seem capable
by itself to explain the presence of a highly polarized core in P1 surrounded
by low polarization wings, unless the PA swings much faster than in optical.
Intrinsic depolarization is most likely required. Analytical striped-wind
emission models suggest possible lower polarization in B, but also predict a
fully unpolarized P1 Petri13a . However recent models, focused on emission in
the wind and outer magnetosphere, based on numerical magnetospheric solution
have shown that the polarization signatures are highly sensitive to the
location and geometry of the emission region Ceritti_Mortier+16a ;
Harding_Kalapotharakos17a . Low integrated polarization suggests that the
emission region should be close to or beyond the Light Cylinder
Harding_Kalapotharakos17a . However, none of the current models include
important physical ingredients: micro-turbulence, which is likely present in
the wind current sheet Cerutti_Philippov+20a , and could lead to significative
depolarization; short time-scale variability which manifests as timing noise
Scott_Finger+03a  and could lead to potential depolarization for long time
integration.

A detailed comparison with previous measures, typically in higher energy
ranges, or with optical data would benefit from a better statistics, and would
require further modeling/extrapolation (to account for changes in the pulse
shape with energy) and goes beyond the scope of this work. Previous measures
in hard X-rays for the PD of the integrated pulsed emission have values
typically higher $\sim$20%-30%. These are well above our IXPE MDP for the
pulsed emission. Our measure is in clear contrast with these high energy PD
estimates. Note however, that the more recent PolarLight narrow band [3-4.5]
keV measures Long_Feng+21a  are consistent with our findings that the PSR
emission is likely unpolarized. A strong decrease in the PD, from the optical
to the soft X-ray (with a possible recovery to large PD in the hard X-ray) is
not expected in existing modeling, which primarily relies on geometry of the
emission zone to determine the polarization (e.g. Harding_Kalapotharakos17a );
as noted above, additional physical effects will be required to accommodate
the IXPE data.

The PWN shows a polarisation pattern that is consistent with a predominantly
toroidal magnetic field, extending well beyond the observed location of the
X-ray torus. For synchrotron radiation this is consistent with general
expectation from MHD modelling of this source Bucciantini_del-Zanna+05a ;
Nakamura_Shibata07a ; Porth_Komissarov+14a ; Olmi_Del-Zanna+16a ;
Bucciantini_Bandiera+17a . We found however that the mostly symmetric PA (i.e.
magnetic field) m pattern is associated with large asymmetries in the PD,
likely indicating variations in the level of turbulence inside the PWN. Such
level of asymmetry is similar to, but stronger than, that seen in the
intensity maps, and reflects a similar trend found in optical polarisation
images Hester08a .

The magnetic axis of the PWN, derived by taking the symmetry axis of the
magnetic field pattern, is found to be $\sim 140^{\circ}$, larger by about 15◦
than estimates based on fitting axisymmetric structure to the torus intensity
Ng_Romani04a . It is also possible to estimate the inclination of the magnetic
axis with respect to the plane of the sky: we found it to be $sim60^{\circ}$,
in agreement with previous estimates. While the average PD $\sim$20% agrees
with previous measures, the PA differs in a statistically significant way from
other estimates, reflecting the spatial variation of the PD, or possible
temporal variability. The spatially resolved PD reaches a maximum of
$\sim$46-50%. This is about two times larger than expected from simple
predictions based on synchrotron turbulent modelling of the Crab Torus and
Inner Ring luminosity profiles, calibrated on the OSO-8 results
Bucciantini_Bandiera+17a . More sophisticate 3D models (lacking however micro-
turbulence) can give PD close to the theoretical maximum $\sim$0.7
Porth_Komissarov+14a , with higher values typically in the South-West region,
but in general the prediction is for polarised patterns quite symmetric with
respect to the nebular axis, unlike what was found. This suggests that the
level and development of turbulence within the nebula, is not as strong as
predicted and much patchier in its spatial distribution. While the lower level
of polarization close to the center of the PWN is easily explained by summed
emission from a wide range of PA in the central resolution elements, the
increase of the PD with distance at the rim of the torus suggests the presence
of a highly ordered magnetic layer at the edge of the torus itself [the ratio
of the energy in the turbulent versus ordered magnetic field components should
be about a factor 2 smaller than in the core of the torus Bandiera_Petruk16a
]. This differs from what is seen in optical where higher polarization is
found in inner features Moran_Shearer+13a , suggesting that optical and X-ray
emitting particles might be accelerated in different locations and sample
different regions of the nebula as previously suggested
Schweizer_Bucciantini+13a . The fact that the PD (which depends on the ratio
of magnetic energy in the turbulent to ordered components) is far more
asymmetric, with respect to the nebular axis, than the intensity (which
depends on the total turbulent plus ordered magnetic energy density) suggests
that the level of turbulence anti-correlates with the strength of the ordered
component of the magnetic field. This is what one would expect if turbulence
was driven by the growth of instabilities, like Rayleigh-Taylor, which are
suppressed by stronger fields Bucciantini_Amato+04a . If this is correct we
should expect that more highly polarized PWNe (less turbulent systems) should
show a stronger toroidal patterns with with smaller degree of Rayleigh-Taylor
induced patchy de-polarisation and intensity enhancement [see e.g. the recent
IXPE observation of the Vela PWN (Fei et al. 2022, submitted to Nature)].

The IXPE polarisation results indicate that present modeling lacks physical
ingredients needed to explain the low pulsar polarisation seen at most phases.
The substantial spatial variation of the PD in the nebula also indicates that
effects are missing even in the most advanced 3D relativistic Magneto-
Hydrodynamical models; MHD turbulence seems likely to be important in both
cases.

Supplementary information Supplementary material includes Figures S1 to S3,
and Tables S1 to S2.

Acknowledgments The Imaging X-ray Polarimetry Explorer (IXPE) is a joint US
and Italian mission. The US contribution is supported by the National
Aeronautics and Space Administration (NASA) and led and managed by its
Marshall Space Flight Center (MSFC), with industry partner Ball Aerospace
(contract NNM15AA18C). The Italian contribution is supported by the Italian
Space Agency (Agenzia Spaziale Italiana, ASI) through contract ASI-
OHBI-2017-12-I.0, agreements ASI-INAF-2017-12-H0 and ASI-INFN-2017.13-H0, and
its Space Science Data Center (SSDC), and by the Istituto Nazionale di
Astrofisica (INAF) and the Istituto Nazionale di Fisica Nucleare (INFN) in
Italy. This research used data products provided by the IXPE Team (MSFC, SSDC,
INAF, and INFN) and distributed with additional software tools by the High-
Energy Astrophysics Science Archive Research Center (HEASARC), at NASA Goddard
Space Flight Center (GSFC). The research at Boston University was supported in
part by National Science Foundation grant AST-2108622. Part of the French
contributions is supported by the Scientific Research National Center (CNRS)
and the French spatial agency (CNES).

Author contributions NB led the data analysis and the writing of the paper.
RF, MB, JR, LP, FM contributed to data analysis and data calibration. NDL, CS,
NO, TK, TM, SG, EW contributed to data analysis and results interpretation.
MCW, MN, SS, EDOW, FX, JH, RWR, PT, AP, HLM contributed to text revision, and
data interpretation. LB and MPR contribute to software development. The
remaining members of the IXPE collaboration contributed to the design of the
mission, to the calibration of the instrument, to define its science case and
to the planning of the observations. All authors provided inputs and comments
on the manuscript. Competing interests Authors declare that they have no
competing interests. Availability of data and materials Data of the Crab
pulsar and nebula observation are available in the HEASARC IXPE Data Archive
(https://heasarc.gsfc.nasa.gov/docs/ixpe/archive/).

Table 1: Global polarisation properties of the PSR+PWN complex. Normalized Stokes parameters, Polarised Degree and Angle for various energy and phase ranges (in brackets the 1$\sigma$ errors). The OP is in the [2-8] keV range. The significance (Sig) is given as the ratio of PD over its 1$\sigma$ error. See Figure 1 Selection | Q/I | U/I | PD[%] | PA[deg] | Sig
---|---|---|---|---|---
PSR+PWN [2-8] keV | 0.177(0.0019) | 0.068(0.0019) | 19.0(0.19) | 145.5(0.29) | 99
PSR+PWN [2-4] keV | 0.168(0.0019) | 0.081(0.0019) | 18.7(0.19) | 147.8(0.29) | 100
PSR+PWN [4-8] keV | 0.199(0.0038) | 0.037(0.0039) | 20.2(0.38) | 140.2(0.55) | 53
PWN (OFF Pulse) | 0.189(0.0036) | 0.073(0.0036) | 20.2(0.36) | 145.6(0.51) | 57
Figure 1: Global Polarisation properties of the Crab PWN$+$PSR complex.
Emission is integrated over a region of 2.5’ radius centered on the PSR.
Normalized Stokes parameters are shown for the total emission, the emission in
the [2-4] keV and [4-8] keV energy bands, and for the OP only, together with
contours of polarised degree (in %) and angle.

Figure 2: omparison of Chandra and IXPE images. Top Left panel: Chandra image
(ObsId235309) (intensity map) of the Crab PWN in the [2-8] keV energy range.
Top Right panel: cartoon of the jet-torus structure indicating the main
features observed in X-rays. Bottom Left panel: Total IXPE count map in the
[2-8] keV energy range. Bottom Right panel: OP (see the Supplementary
Materials for the definition of its phase range) only IXPE count map in the
[2-8] keV energy range. The white circular region of 20” radius is the one
used to do the phase-resolved polarimetry of the PSR. The black box represents
the region corresponding to the Chandra image.

Figure 3: Polarization properties of the Crab PSR. Normalized Stokes
Parameters with their 1$\sigma$ errors for the OP subtracted emission of the
Crab PSR in the [2-8] keV energy bands (blue crosses), overlaid with the phase
bins of interest: OP (red), P1 and P2 centers (green), left wings (yellow) and
right wings(blue), and B (orange) (see the Supplementary Materials for the
definition of the various phase ranges). Overlaid is the OP subtracted PSR
light curve (counts) in the [2-8] keV energy band, normalised and rescaled to
the range of the y-axis (grey solid line).

Figure 4: Polarized Structure of the Crab nebula. Right panel: Total intensity
map of the Crab PWN$+$PSR complex in the [2-8] keV energy band, overlaid with
the reconstructed polarisation direction (magnetic field). Left panel: map of
the polarized fraction (PD/100) cut above 5$\sigma$ confidence level. The gray
dashed line id the nebular axis inferred from X-ray intensity maps
Ng_Romani04a . Overlaid are the Sky directions for ease of reference.

## References

  *   * (1) Rees, M.J., Gunn, J.E.: The origin of the magnetic field and relativistic particles in the Crab Nebula. MNRAS 167, 1–12 (1974)
  * (2) Gaensler, B.M., Slane, P.O.: The Evolution and Structure of Pulsar Wind Nebulae. ARA&A 44, 17–47 (2006) arXiv:astro-ph/0601081. https://doi.org/10.1146/annurev.astro.44.051905.092528
  * (3) Hester, J.J.: The Crab Nebula: An Astrophysical Chimera. ARA&A 46, 127–155 (2008). https://doi.org/10.1146/annurev.astro.45.051806.110608
  * (4) Bühler, R., Blandford, R.: The surprising Crab pulsar and its nebula: a review. Reports on Progress in Physics 77(6), 066901 (2014) arXiv:1309.7046 [astro-ph.HE]. https://doi.org/%****␣sn-article.bbl␣Line␣100␣****10.1088/0034-4885/77/6/066901
  * (5) Weisskopf, M.C., Silver, E.H., Kestenbaum, H.L., Long, K.S., Novick, R.: A precision measurement of the X-ray polarization of the Crab Nebula without pulsar contamination. ApJL 220, 117–121 (1978). https://doi.org/10.1086/182648
  * (6) Forot, M., Laurent, P., Grenier, I.A., Gouiffès, C., Lebrun, F.: Polarization of the Crab Pulsar and Nebula as Observed by the INTEGRAL/IBIS Telescope. ApJL 688, 29–32 (2008) arXiv:0809.1292. https://doi.org/10.1086/593974
  * (7) Moran, P., Shearer, A., Mignani, R.P., Słowikowska, A., De Luca, A., Gouiffès, C., Laurent, P.: Optical polarimetry of the inner Crab nebula and pulsar. MNRAS 433, 2564–2575 (2013) arXiv:1305.6824 [astro-ph.HE]. https://doi.org/10.1093/mnras/stt931
  * (8) Chauvin, M., Roques, J.P., Clark, D.J., Jourdain, E.: Polarimetry in the Hard X-Ray Domain with INTEGRAL SPI. ApJ 769(2), 137 (2013) arXiv:1305.0802 [astro-ph.IM]. https://doi.org/10.1088/0004-637X/769/2/137
  * (9) Chauvin, M., Florén, H.-G., Friis, M., Jackson, M., Kamae, T., Kataoka, J., Kawano, T., Kiss, M., Mikhalev, V., Mizuno, T., Ohashi, N., Stana, T., Tajima, H., Takahashi, H., Uchida, N., Pearce, M.: Shedding new light on the Crab with polarized X-rays. Scientific Reports 7, 7816 (2017) arXiv:1706.09203 [astro-ph.HE]. https://doi.org/10.1038/s41598-017-07390-7
  * (10) Vadawale, S.V., Chattopadhyay, T., Mithun, N.P.S., Rao, A.R., Bhattacharya, D., Vibhute, A., Bhalerao, V.B., Dewangan, G.C., Misra, R., Paul, B., Basu, A., Joshi, B.C., Sreekumar, S., Samuel, E., Priya, P., Vinod, P., Seetha, S.: Phase-resolved X-ray polarimetry of the Crab pulsar with the AstroSat CZT Imager. Nature Astronomy 2, 50–55 (2018). https://doi.org/10.1038/s41550-017-0293-z
  * (11) Feng, H., Li, H., Long, X., Bellazzini, R., Costa, E., Wu, Q., Huang, J., Jiang, W., Minuti, M., Wang, W., Xu, R., Yang, D., Baldini, L., Citraro, S., Nasimi, H., Soffitta, P., Muleri, F., Jung, A., Yu, J., Jin, G., Zeng, M., An, P., Brez, A., Latronico, L., Sgro, C., Spandre, G., Pinchera, M.: Re-detection and a possible time variation of soft X-ray polarization from the Crab. Nature Astronomy 4, 511–516 (2020) arXiv:2011.05487 [astro-ph.HE]. https://doi.org/10.1038/s41550-020-1088-1
  * (12) Long, X., Feng, H., Li, H., Zhu, J., Wu, Q., Huang, J., Minuti, M., Jiang, W., Wang, W., Xu, R., Costa, E., Yang, D., Citraro, S., Nasimi, H., Yu, J., Jin, G., Zeng, M., An, P., Baldini, L., Bellazzini, R., Brez, A., Latronico, L., Sgrò, C., Spandre, G., Pinchera, M., Muleri, F., Soffitta, P.: X-Ray Polarimetry of the Crab Nebula with PolarLight: Polarization Recovery after the Glitch and a Secular Position Angle Variation. ApJL 912(2), 28 (2021) arXiv:2104.11391 [astro-ph.HE]. https://doi.org/10.3847/2041-8213/abfb00
  * (13) Li, H.-C., Produit, N., Zhang, S.-N., Kole, M., Sun, J.-C., Ge, M.-Y., De Angelis, N., Hulsman, J., Li, Z.-H., Song, L.-M., Tymieniecka, T., Wu, B.-B., Wu, X., Wang, Y.-H., Xiong, S.-L., Zhang, Y.-J., Zhao, Y., Zheng, S.-J.: Gamma-ray polarimetry of the Crab pulsar observed by POLAR. MNRAS 512(2), 2827–2840 (2022) arXiv:2202.10877 [astro-ph.HE]. https://doi.org/10.1093/mnras/stac522
  * (14) Trimble, V.: The Distance to the Crab Nebula and NP 0532. Publications of the Astronomical Society of the Pacific 85(507), 579 (1973). https://doi.org/10.1086/129507
  * (15) Kaaret, P., Marshall, H.L., Aldcroft, T.L., Graessle, D.E., Karovska, M., Murray, S.S., Rots, A.H., Schulz, N.S., Seward, F.D.: Chandra Observations of the Young Pulsar PSR B0540-69. ApJ 546(2), 1159–1167 (2001) arXiv:astro-ph/0008388 [astro-ph]. https://doi.org/10.1086/318287
  * (16) Massaro, E., Campana, R., Cusumano, G., Mineo, T.: The optical to $\gamma$-ray emission of the Crab pulsar: a multicomponent model. A&A 459(3), 859–870 (2006) arXiv:astro-ph/0607410 [astro-ph]. https://doi.org/10.1051/0004-6361:20065118
  * (17) Weisskopf, M.C., Tennant, A.F., Yakovlev, D.G., Harding, A., Zavlin, V.E., O’Dell, S.L., Elsner, R.F., Becker, W.: Chandra Phase-resolved X-Ray Spectroscopy of the Crab Pulsar. ApJ 743(2), 139 (2011) arXiv:1106.3270 [astro-ph.GA]. https://doi.org/10.1088/0004-637X/743/2/139
  * (18) Ge, M.Y., Lu, F.J., Qu, J.L., Zheng, S.J., Chen, Y., Han, D.W.: X-Ray Phase-resolved Spectroscopy of PSRs B0531+21, B1509-58, and B0540-69 with RXTE. ApJ Supp. 199, 32 (2012) arXiv:1204.2199 [astro-ph.HE]. https://doi.org/%****␣sn-article.bbl␣Line␣450␣****10.1088/0067-0049/199/2/32
  * (19) Vivekanand, M.: Phase-resolved spectrum of the Crab pulsar from NICER. A&A 649, 140 (2021) arXiv:2102.04675 [astro-ph.HE]. https://doi.org/10.1051/0004-6361/202140358
  * (20) Słowikowska, A., Kanbach, G., Kramer, M., Stefanescu, A.: Optical polarization of the Crab pulsar: precision measurements and comparison to the radio emission. MNRAS 397(1), 103–123 (2009) arXiv:0901.4559 [astro-ph.SR]. https://doi.org/10.1111/j.1365-2966.2009.14935.x
  * (21) Kargaltsev, O., Pavlov, G.G.: Pulsar Wind Nebulae in the Chandra Era. In: C. Bassa, Z. Wang, A. Cumming, & V. M. Kaspi (ed.) 40 Years of Pulsars: Millisecond Pulsars, Magnetars and More. American Institute of Physics Conference Series, vol. 983, pp. 171–185 (2008). https://doi.org/%****␣sn-article.bbl␣Line␣500␣****10.1063/1.2900138
  * (22) Mori, K., Burrows, D.N., Hester, J.J., Pavlov, G.G., Shibata, S., Tsunemi, H.: Spatial Variation of the X-Ray Spectrum of the Crab Nebula. ApJ 609, 186–193 (2004) arXiv:astro-ph/0403287. https://doi.org/10.1086/421011
  * (23) Hester, J.J., Scowen, P.A., Sankrit, R., Burrows, C.J., Gallagher, J.S. III, Holtzman, J.A., Watson, A., Trauger, J.T., Ballester, G.E., Casertano, S., Clarke, J.T., Crisp, D., Evans, R.W., Griffiths, R.E., Hoessel, J.G., Krist, J., Lynds, R., Mould, J.R., O’Neil, E.J. Jr., Stapelfeldt, K.R., Westphal, J.A.: WFPC2 Studies of the Crab Nebula. I. HST and ROSAT Imaging of the Synchrotron Nebula. ApJ 448, 240 (1995). https://doi.org/10.1086/175956
  * (24) Weisskopf, M.C., Hester, J.J., Tennant, A.F., Elsner, R.F., Schulz, N.S., Marshall, H.L., Karovska, M., Nichols, J.S., Swartz, D.A., Kolodziejczak, J.J., O’Dell, S.L.: Discovery of Spatial and Spectral Structure in the X-Ray Emission from the Crab Nebula. ApJL 536, 81–84 (2000) arXiv:astro-ph/0003216. https://doi.org/10.1086/312733
  * (25) Aumont, J., Conversi, L., Thum, C., Wiesemeyer, H., Falgarone, E., Macías-Pérez, J.F., Piacentini, F., Pointecouteau, E., Ponthieu, N., Puget, J.L., Rosset, C., Tauber, J.A., Tristram, M.: Measurement of the Crab nebula polarization at 90 GHz as a calibrator for CMB experiments. A&A 514, 70 (2010) arXiv:0912.1751. https://doi.org/10.1051/0004-6361/200913834
  * (26) Moran, P., Shearer, A., Mignani, R.P., Słowikowska, A., De Luca, A., Gouiffès, C., Laurent, P.: Optical polarimetry of the inner Crab nebula and pulsar. MNRAS 433, 2564–2575 (2013) arXiv:1305.6824 [astro-ph.HE]. https://doi.org/10.1093/mnras/stt931
  * (27) Weisskopf, M.: The Imaging X-ray Polarimetry Explorer (IXPE) Mission Overview. In: AAS/High Energy Astrophysics Division. AAS/High Energy Astrophysics Division, vol. 54, pp. 301–01 (2022)
  * (28) Weisskopf, M.C., Soffitta, P., Baldini, L., Ramsey, B.D., O’Dell, S.L., Romani, R.W., Matt, G., Deininger, W.D., Baumgartner, W.H., Bellazzini, R., Costa, E., Kolodziejczak, J.J., Latronico, L., Marshall, H.L., Muleri, F., Bongiorno, S.D., Tennant, A., Bucciantini, N., Dovciak, M., Marin, F., Marscher, A., Poutanen, J., Slane, P., Turolla, R., Kalinowski, W., Marco, A.D., Fabiani, S., Minuti, M., Monaca, F.L., Pinchera, M., Rankin, J., Sgrò, C., Trois, A., Xie, F., Alexander, C., Allen, D.Z., Amici, F., Andersen, J., Antonelli, A., Antoniak, S., Attiná, P., Barbanera, M., Bachetti, M., Baggett, R.M., Bladt, J., Brez, A., Bonino, R., Boree, C., Borotto, F., Breeding, S., Brienza, D., Bygott, H.K., Caporale, C., Cardelli, C., Carpentiero, R., Castellano, S., Castronuovo, M., Cavalli, L., Cavazzuti, E., Ceccanti, M., Centrone, M., Citraro, S., D’Amico, F., D’Alba, E., Gesu, L.D., Monte, E.D., Dietz, K.L., Lalla, N.D., Persio, G.D., Dolan, D., Donnarumma, I., Evangelista, Y., Ferrant, K., Ferrazzoli, R., Ferrie, M., Footdale, J., Forsyth, B., Foster, M., Garelick, B., Gunji, S., Gurnee, E., Head, M., Hibbard, G., Johnson, S., Kelly, E., Kilaru, K., Lefevre, C., Roy, S.L., Loffredo, P., Lorenzi, P., Lucchesi, L., Maddox, T., Magazzu, G., Maldera, S., Manfreda, A., Mangraviti, E., Marengo, M., Marrocchesi, A., Massaro, F., Mauger, D., McCracken, J., McEachen, M., Mize, R., Mereu, P., Mitchell, S., Mitsuishi, I., Morbidini, A., Mosti, F., Nasimi, H., Negri, B., Negro, M., Nguyen, T., Nitschke, I., Nuti, A., Onizuka, M., Oppedisano, C., Orsini, L., Osborne, D., Pacheco, R., Paggi, A., Painter, W., Pavelitz, S.D., Pentz, C., Piazzolla, R., Perri, M., Pesce-Rollins, M., Peterson, C., Pilia, M., Profeti, A., Puccetti, S., Ranganathan, J., Ratheesh, A., Reedy, L., Root, N., Rubini, A., Ruswick, S., Sanchez, J., Sarra, P., Santoli, F., Scalise, E., Sciortino, A., Schroeder, C., Seek, T., Sosdian, K., Spandre, G., Speegle, C.O., Tamagawa, T., Tardiola, M., Tobia, A., Thomas, N.E., Valerie, R., Vimercati, M., Walden, A.L., Weddendorf, B., Wedmore, J., Welch, D., Zanetti, D., Zanetti, F.: Imaging X-ray Polarimetry Explorer: prelaunch. Journal of Astronomical Telescopes, Instruments, and Systems 8(2), 1–28 (2022). https://doi.org/10.1117/1.JATIS.8.2.026002
  * (29) Pesce-Rollins, M., Lalla, N.D., Omodei, N., Baldini, L.: An observation-simulation and analysis framework for the Imaging X-ray Polarimetry Explorer (IXPE). Nuclear Instruments and Methods in Physics Research A 936, 224–226 (2019). https://doi.org/10.1016/j.nima.2018.10.041
  * (30) Baldini, L., Bucciantini, N., Di Lalla, N., Ehlert, S.R., Manfreda, A., Omodei, N., Pesce-Rollins, M., Sgrò, C.: ixpeobssim: a Simulation and Analysis Framework for the Imaging X-ray Polarimetry Explorer. arXiv e-prints, 2203–06384 (2022) arXiv:2203.06384 [astro-ph.IM]
  * (31) Tennant, A.F., Becker, W., Juda, M., Elsner, R.F., Kolodziejczak, J.J., Murray, S.S., O’Dell, S.L., Paerels, F., Swartz, D.A., Shibazaki, N., Weisskopf, M.C.: Discovery of X-Ray Emission from the Crab Pulsar at Pulse Minimum. ApJL 554(2), 173–176 (2001). https://doi.org/10.1086/321718
  * (32) Ng, C.-Y., Romani, R.W.: Fitting Pulsar Wind Tori. ApJ 601, 479–484 (2004) astro-ph/0310155. https://doi.org/10.1086/380486
  * (33) Schweizer, T., Bucciantini, N., Idec, W., Nilsson, K., Tennant, A., Weisskopf, M.C., Zanin, R.: Characterization of the optical and X-ray properties of the north-western wisps in the Crab nebula. MNRAS 433, 3325–3335 (2013) arXiv:1301.1321 [astro-ph.HE]. https://doi.org/10.1093/mnras/stt995
  * (34) Dyks, J., Harding, A.K.: Rotational Sweepback of Magnetic Field Lines in Geometric Models of Pulsar Radio Emission. ApJ 614(2), 869–880 (2004) arXiv:astro-ph/0402507 [astro-ph]. https://doi.org/10.1086/423707
  * (35) Pétri, J., Kirk, J.G.: The Polarization of High-Energy Pulsar Radiation in the Striped Wind Model. ApJL 627(1), 37–40 (2005) arXiv:astro-ph/0505427 [astro-ph]. https://doi.org/10.1086/431973
  * (36) Pétri, J.: Phase-resolved polarization properties of the pulsar striped wind synchrotron emission. MNRAS 434(3), 2636–2644 (2013) arXiv:1308.0973 [astro-ph.HE]. https://doi.org/10.1093/mnras/stt1214
  * (37) Cerutti, B., Mortier, J., Philippov, A.A.: Polarized synchrotron emission from the equatorial current sheet in gamma-ray pulsars. MNRAS 463(1), 89–93 (2016) arXiv:1609.00021 [astro-ph.HE]. https://doi.org/%****␣sn-article.bbl␣Line␣975␣****10.1093/mnrasl/slw162
  * (38) Harding, A.K., Kalapotharakos, C.: Multiwavelength Polarization of Rotation-powered Pulsars. ApJ 840(2), 73 (2017) arXiv:1704.06183 [astro-ph.HE]. https://doi.org/10.3847/1538-4357/aa6ead
  * (39) Cerutti, B., Philippov, A.A., Dubus, G.: Dissipation of the striped pulsar wind and non-thermal particle acceleration: 3D PIC simulations. A&A 642, 204 (2020) arXiv:2008.11462 [astro-ph.HE]. https://doi.org/10.1051/0004-6361/202038618
  * (40) Scott, D.M., Finger, M.H., Wilson, C.A.: Characterization of the timing noise of the Crab pulsar. MNRAS 344(2), 412–430 (2003) arXiv:astro-ph/0303582 [astro-ph]. https://doi.org/10.1046/j.1365-8711.2003.06825.x
  * (41) Bucciantini, N., del Zanna, L., Amato, E., Volpi, D.: Polarization in the inner region of pulsar wind nebulae. A&A 443, 519–524 (2005) arXiv:astro-ph/0508212. https://doi.org/10.1051/0004-6361:20053667
  * (42) Nakamura, Y., Shibata, S.: Polarization of the Crab Nebula with disordered magnetic components. MNRAS 381, 1489–1498 (2007) arXiv:astro-ph/0702512. https://doi.org/10.1111/j.1365-2966.2007.12172.x
  * (43) Porth, O., Komissarov, S.S., Keppens, R.: Three-dimensional magnetohydrodynamic simulations of the Crab nebula. MNRAS 438, 278–306 (2014) arXiv:1310.2531 [astro-ph.HE]. https://doi.org/10.1093/mnras/stt2176
  * (44) Olmi, B., Del Zanna, L., Amato, E., Bucciantini, N., Mignone, A.: Multi-D magnetohydrodynamic modelling of pulsar wind nebulae: recent progress and open questions. Journal of Plasma Physics 82(6), 635820601 (2016) arXiv:1610.07956 [astro-ph.HE]. https://doi.org/10.1017/S0022377816000957
  * (45) Bucciantini, N., Bandiera, R., Olmi, B., Del Zanna, L.: Modeling the effect of small-scale magnetic turbulence on the X-ray properties of Pulsar Wind Nebulae. MNRAS 470, 4066–4074 (2017) arXiv:1704.06546 [astro-ph.HE]. https://doi.org/10.1093/mnras/stx993
  * (46) Bandiera, R., Petruk, O.: Radio polarization maps of shell-type supernova remnants - I. Effects of a random magnetic field component and thin-shell models. MNRAS 459, 178–198 (2016) arXiv:1603.01199 [astro-ph.HE]. https://doi.org/10.1093/mnras/stw551
  * (47) Bucciantini, N., Amato, E., Bandiera, R., Blondin, J.M., Del Zanna, L.: Magnetic Rayleigh-Taylor instability for Pulsar Wind Nebulae in expanding Supernova Remnants. A&A 423, 253–265 (2004) arXiv:astro-ph/0405276. https://doi.org/10.1051/0004-6361:20040360
  * (48) Ferrazzoli, R., Muleri, F., Lefevre, C., Morbidini, A., Amici, F., Brienza, D., Costa, E., Del Monte, E., Di Marco, A., Di Persio, G., Donnarumma, I., Fabiani, S., La Monaca, F., Loffredo, P., Maiolo, L., Maita, F., Piazzolla, R., Ramsey, B., Rankin, J., Ratheesh, A., Rubini, A., Sarra, P., Soffitta, P., Tobia, A., Xie, F.: In-flight calibration system of imaging x-ray polarimetry explorer. Journal of Astronomical Telescopes, Instruments, and Systems 6, 048002 (2020) arXiv:2010.14185 [astro-ph.IM]. https://doi.org/10.1117/1.JATIS.6.4.048002
  * (49) Lyne, A.G., Pritchard, R.S., Graham-Smith, F.: Twenty-Three Years of Crab Pulsar Rotational History. MNRAS 265, 1003 (1993)
  * (50) Bachetti, M.: HENDRICS: High ENergy Data Reduction Interface from the Command Shell. Astrophysics Source Code Library, record ascl:1805.019 (2018)
  * (51) Hobbs, G.B., Edwards, R.T., Manchester, R.N.: TEMPO2, a new pulsar-timing package - I. An overview. MNRAS 369(2), 655–672 (2006) arXiv:astro-ph/0603381 [astro-ph]. https://doi.org/10.1111/j.1365-2966.2006.10302.x
  * (52) Luo, J., Ransom, S., Demorest, P., Ray, P.S., Archibald, A., Kerr, M., Jennings, R.J., Bachetti, M., van Haasteren, R., Champagne, C.A., Colen, J., Phillips, C., Zimmerman, J., Stovall, K., Lam, M.T., Jenet, F.A.: PINT: A Modern Software Package for Pulsar Timing. ApJ 911(1), 45 (2021) arXiv:2012.00074 [astro-ph.IM]. https://doi.org/10.3847/1538-4357/abe62f
  * (53) Kuiper, L., Hermsen, W., Walter, R., Foschini, L.: Absolute timing with IBIS, SPI and JEM-X aboard INTEGRAL. Crab main-pulse arrival times in radio, X-rays and high-energy gamma -rays. A&A 411, 31–36 (2003) arXiv:astro-ph/0309178 [astro-ph]. https://doi.org/10.1051/0004-6361:20031353

## Methods

###  3.1 Observations and Data Analysis

The Imaging X-ray Polarimetry Explorer (IXPE), is a NASA mission in
partnership with the Italian space agency (ASI), launched in December 9 2021.
As described in detail elsewhere [Weisskopf22d ; Weisskopf_Soffitta+22x  and
references therein], the IXPE Observatory includes three identical X-ray
telescopes (DUs), each comprising a Wolter-I X-ray mirror assembly (NASA-
provided) with with angular resolution (half-power diameter) of 19” (DU1), 26”
(DU2), 28” (DU3), and a polarisation-sensitive pixelated detector (GDP, ASI-
provided), with a typical energy-dependent dead-time of $\sim$1.1 ms. This
allows one to measure the energy, arrival direction, arrival time and linear
polarization of the detected X-ray signal, which are all reconstructed from
the photo-electron track shape using moment analysis. The IXPE energy range is
the [2-8] keV band, with a total effective area of 590 cm2 at 4.5 keV. The
modulation factor (the amplitude of the modulation of the reconstructed photo-
electrons angle distribution for a 100% polarised source) ranges from
$\sim$15% at 2 keV up to $\sim$60% at 8 keV Weisskopf_Soffitta+22x .

The Crab PWN and PSR were observed twice: the first time from 2022-02-21
UTC16:13:32 to 2022-02-22 UTC18:46:37, the second from 2022-03-07 UTC00:14:20
to 2022-03-08 UTC02:40:02, for a total 92363 seconds of ONTIME exposure (the
total exposure as obtained from the sum of the good time intervals), and 85062
seconds of total LIVETIME (the total amount of time the CCD was actively
observing a source; it excludes the time it takes to transfer charge from the
image region to the frame store region).

The polarisation analysis was performed on publicly available level 2 event
list files. These were corrected to account for a few calibration issues that
have emerged during flight operations. The World Coordinate System (WCS) was
corrected to account for the small offset among the various units, registering
the pointing solution in order to center the intensity peak of each unit on
the PSR position at RA = 5h:34m:31.86s, DEC = 22:00:51.3. The time dependent
charge-to-energy conversion was reconstructed for each units using the two
onboard calibration sources at 1.7 keV (Si K$\alpha$) and 5.9 keV
(55Fe$\rightarrow$ 55Mg with following K$\alpha$ emission)
Ferrazzoli_Muleri+20a . Spurious offsets in the pointing solution (aspect
solution), associated to the switch between different star trackers during
orbit, were identified looking at the time variations of the count rate in a
set of background sky regions, and later filtered out by removing the affected
time intervals. This results in a loss of counts smaller than 2% and an
effective new on-source time of $\sim$91ks. At the time of the Crab
observation, the optical axis of the mirror system relative to the star
trackers had not yet been accurately determined and was not yet compensated
(by offset pointing). Consequently, the Crab PSR was about 2.8’ off-axis with
respect to the mirror system. This precluded accurate computation of Image
Response Functions (IRFs)-energy dependent vignetting and energy dependent
exposure maps—necessary for a correct spectral analysis (the mirror-
system–star-tracker offset has now been accurately determined and compensated,
such that future observations will place the image close to the mirror-system
optical axis. Furthermore, it may be possible, in the future, to recalibrate
the IRFs), and for computing correct counts rates. However this should not
affect polarisation measures which comes from flux ratios among Stokes
parameters.

###  3.2 Timing Analysis

We initially used the Jodrell Bank Crab PSR Monthly ephemeris
(http://www.jb.man.ac.uk/pulsar/crab.html) Lyne_Pritchard+93a  to calculate
the pulse phase of each photon. However, the time span of the two IXPE
observations requires two separate JB solutions, and the alignment between the
arrival time of the pulse in the two observations using these ephemerides is
visibly off ($\sim$0.02 in phase). Therefore, we determined a new ephemeris by
using the X-ray data alone, as follows. As a starting point, we used the JB
montly ephemeris of February in CGRO format
(https://www.jb.man.ac.uk/$\sim$pulsar/crab/CGRO_format.html) , but modifying
the frequency and derivatives in order for them to refer to an epoch between
the two observations:

$\displaystyle\nu_{\rm new}=$ $\displaystyle\nu_{\rm old}+\dot{\nu}_{\rm
old}(T_{\rm new}-T_{\rm old})+0.5\ddot{\nu}_{\rm old}(T_{\rm new}-T_{\rm
old})^{2}$ (1) $\displaystyle\dot{\nu}_{\rm new}=$
$\displaystyle\ddot{\nu}_{\rm old}(T_{\rm new}-T_{\rm old})$ (2)

where $\nu_{\rm old}$, $\dot{\nu}_{\rm old}$, and $\ddot{\nu}_{\rm old}$ are
the frequency, its first and second time derivative at $T_{\rm old}$, while
$\nu_{\rm new}$, $\dot{\nu}_{\rm new}$d are the frequency and its first time
derivative at $T_{\rm new}$.

Then, we calculated times of arrival (TOAs) of the pulse using the
HENphaseogram tool distributed with HENDRICS
(https://hendrics.stingray.science) Bachetti08a . This tool folds the data in
small fractions of the observation, creating a series of pulse profiles. Then,
it calculates the misalignment between each of these profiles and a smoothed
version of the folded profile from the whole observation, and transforms this
misalignment into a TOA. The TOA refers to the maximum of the reference
profile. As an output, the tool produces a parameter file and a TOA file in
Tempo2 format Hobbs_Edwards+06a . We loaded these approximate parameters and
TOAs in the pintk graphical interface to PINT (https://nanograv-
pint.readthedocs.io) Luo_Ransom+21a , and fitted a new spin down solution that
aligned the TOAs of the two observations. Then, we went back and calculated
new TOAs, this time using the improved timing solution and, consequently, the
better resolved total pulse profile that the solution provided, and fitted
these TOAs again with PINT. We stopped this iterative procedure when the
improvement in the fitting through PINT was smaller than the uncertainties.
Finally, we calculated the closest TOA to the epoch chosen for the timing
solution, and referred the frequency and derivatives to this time, using the
above equation, in order to have a single number for the reference TOA and the
PEPOCH of the timing solution for convenience. The new ephemeris is reported
in Table S1 of the Supplementary Material. Note that, having chosen the
reference time in between the February and March observations, the
determination of the second time derivative of the period is highly uncertain,
but on the other hand the correction due to the second time derivative is not
significative, and can potentially be neglected. Note moreover that the
absolute time alignment of the pulse profile is not necessary for the analysis
in this Paper. However we did verify that the X-ray pulse leads the radio
pulse (as provided by the JB ephemeris) by $\sim$300 $\mu$s, consistent with
past observations from other missions (Kuiper_Hermsen+03a, , e.g.). The Crab
pulsar is a young one and his timing can be quite noisy, to the paint that
ephemeris cannot be extended beyond their range of validity.

###  3.3 Polarization Analysis

The polarisation analysis of both the PWN and the PSR was performed using the
IXPE collaboration software ixpeobssim V26.3.2. ixpeobssim has been designed
to act both as a simulation software and for data reduction. We opted for a
more robust and established unweighted analysis, limited to the [2-8] keV
energy range. Phase folding was performed with xpphase, while event selection
was done using the xpselect tool. Polarization was computed with the xpbin
tool and PCUBE and PMAPCUBE methods.

Being Stokes parameters additive, their values relative to each event,
calibrated for the known spurious modulation of the instrument, and corrected
for the modulation factor Weisskopf_Soffitta+22x  , are summed over the
spatial region and/or phase range of interest. ixpeobssim can compute all
polarisation relevant quantities including their error and the significance.
Deformation of the phase-resolved light curve (pulse shape) due to dead time,
with respect to the dead-time corrected one, is estimated to be less than 3%
[difference between the dead time correction when the count rate is at maximum
(P1) and the one when to the count rate at minimum (OP)], and was thus
ignored. The net PSR’s Stokes parameters in a given phase bin are obtained as
follows: the I, Q and U values for the OP are subtracted from those of the
pulse phase bin of interest (after scaling then to the phase bin width) to
give a net, OP-subtracted, I, Q and U. Figure S2 of the Supplementary Material
shows the maps of normalized Stokes parameters, of the total PWN plus PSR
emission. The contribution of the lowly polarised PSR is hardly visible. We
verified that the pattern, apart from counting noise, is the same if one
considers just the emission in the OP phase range. While the U/I map shows a
high level of symmetry with respect to the direction of the nebular axis
inferred from the intensity maps Ng_Romani04a , the one for Q/I is far more
asymmetric (its symmetry axis is more aligned with the North-South direction).
The inclination of the magnetic axis of the nebula was derived by fitting
ellipses to the internal magnetic field structure as displayed in Figure 4.

###  3.4 CHANDRA Observation

The Crab PWN was observed by CHANDRA (OsbId23539) starting on 2022-03-15 at
11:32:23 UTC and ending on 2022-03-15 at 15:14:17 UTC, for a total of $\sim$
104 seconds; due to telemetry saturation from the bright source, the effective
exposure time was 1331 seconds. Data were processed with the CIAO package 4.14
using CALDB 4.9.7, with the chandra repro mode=h tool using default settings,
and the [2-8] keV image was done with the fluximage tool, and later smoothed
with a gaussian kernel using aconvolve kernelspec=’lib:gaus(2,5,1,1,1)’

## Supplementary Material

Figure S1. Comparison of the polarisation measured between IXPE and OSO-8 nearly 50 years before, both PSR subtracted, in a region of 110arcsec centered on the PSR, and covering the entire PWN. Contours are at 1$\sigma$ confidence (67%). Spectral response of OSO-8 was strongly peaked in a narrow energy band, and for this reason we selected in the IXPE analysis only the events in two narrow energy bands bracketing the ones of OSO-8. The comparison shows that the polarisation degree measured by OSO-8 are fully in agreement with those measured by IXPE, whereas a rotation of the polarisation angle of about 10∘ is detected at more than 2$\sigma$ confidence for the lower energy band, and at about 1$\sigma$ level for the higher energy band. Figure S2. Maps of the total PWN & PSR normalised Stokes Parameters Q/I and U/I, in the [2-8] keV energy band, smoothed with a Gaussian kernel, and limited to pixels having intensity above 0.3% of the maximum.  Figure S3. Left panel: map of the significance (ratio of polarised degree over its 1$\sigma$ uncertainty) in the [2-8] keV energy band, smoothed with a Gaussian kernel, and limited to pixels having intensity above 0.3% of the maximum. Right panel: map of the Minimum Detectable Polarisation (MDP) at 99% confidence level in the same energy band. Table S1 PSR ephemeris as determined from IXPE events time of arrival. F0 is the period, F1 the period time derivative, F2 the period second time derivative, all at PEPOCH. In brackets the 1$\sigma$ errors on the last digit.  PEPOCH (MJD) | 59639.358974437835272
---|---
F0 (s) | 29.5866194471(4)
F1 | -3.67445(8)e-10
F2 (s-1) | -4(4) e-20
Table S2. OP subtracted (and OP) polarisation properties of various phase bins (a phase offset has been added to shift P1 at phase 0.13, for ease of visualization). In brackets the 1$\sigma$ errors. See also Figure 3. Phase Bin | Phase Range | Q/I | U/I
---|---|---|---
P1 left wing | [0.085-0.120] | -0.068(0.033) | -0.007(0.033)
P1 center | [0.120-0.140] | -0.132(0.025) | -0.079(0.025)
P1 right wing | [0.140-0.175] | 0.044(0.038) | -0.003(0.038)
P2 left wing | [0.460-0.515] | -0.027(0.033) | -0.023(0.033)
P2 center | [0.515-0.545] | -0.013(0.030) | -0.06(0.030)
P2 right wing | [0.545-0.600] | 0.008(0.047) | -0.045(0.047)
Bridge | [0.200-0.400] | 0.056(0.081) | 0.137(0.081)
Off-Pulse | [0.700-1.000] | -0.01606(0.008) | -0.241(0.008)


# A note on hospital financing:
local financing vs. central financing

Raffaele Mosca

###### Abstract

This note tries to study how hospital behaviors, with reference to
interhospital collaboration or competition, could be affected by hospital
financing systems.

For that this note simulates two scenarios which start with the following
baseline scenario: a State, with a set of hospitals, each with all types of
wards at a basic level. The evolution of this baseline scenario consists in
the evolution of hospitals, that is, in the possibility of hospitals to make
some of their wards excel. The State has a budget, for the evolution of this
baseline scenario, which can be used by two financing systems: either by a
”local financing”, i.e., by splitting the budget among the hospitals so that
each hospital is managing its own portion of the budget by pursuing the
individual benefit, or by a ”central financing”, i.e., by not splitting the
budget among the hospitals so that the State is the sole manager of the
budget, by pursuing the benefit of the whole community.

The conclusions seem to be that: in the local financing system hospitals tend
to diversify their excellences, while in the central financing system the
State tends to create poles of excellence.

Dipartimento di Economia, Universitá degli Studi ”G. d’Annunzio”, Pescara
65127, Italy.
<EMAIL_ADDRESS>
##  1 Introduction

This note is inspired to reference [7] where the authors study interhospital
collaboration and competition. In particular, interhospital collaboration
seems to rely on patient transfers, while interhospital competition seems to
rely on ability to attract patients. Such aspects in turn seem to rely on the
”level difference”, i.e. on the suitability difference, between pairs of
hospitals.

Then this note tries to study how hospital behaviors, with reference to such
aspects, could be affected by hospital financing system: in fact, the
assumption is that the ”level difference” between pairs of hospitals in turn
relies on the health policy of making hospital wards excel, which finally
relies on hospital financing systems.

For that this note simulates two scenarios which start with the following
baseline scenario: a State, with a set of hospitals, each with all types of
wards at a basic level. The evolution of this baseline scenario consists in
the evolution of hospitals, that is, in the possibility of hospitals to make
some of their wards excel. The State has a budget, for the evolution of this
baseline scenario, which can be used by two financing systems: either by a
”local financing”, i.e., by splitting the budget among the hospitals so that
each hospital is managing its own portion of the budget by pursuing the
individual benefit, or by a ”central financing”, i.e., by not splitting the
budget among the hospitals so that the State is the sole manager of the
budget, by pursuing the benefit of the whole community.

In details:

\- Section 2 introduces the above baseline scenario and the above financing
systems (local financing, central financing) according to which this baseline
scenario can evolve;

\- Section 3 studies the local financing system as a Game Theory problem;

\- Section 4 studies the central financing system as a Mathematical
Programming problem.

The conclusions seem to be that: in the local financing system hospitals tend
to diversify their excellences, while in the central financing system the
State tends to create poles of excellence.

Let us mention that the question ”centralization or decentralization” in
health policy is treated in [10] (cf. Chapter 3) in terms of decision-making
and in more general terms, as well as in different papers [2, 4, 5, 6, 11, 13]
in different terms.

Let us observe finally that similar conclusions could be reached for similar
contexts, e.g., for university financing.

##  2 Baseline scenario

Assume that the baseline scenario is defined as follows:

  * (1.1)

the context is a $State$;

  * (1.2)

there is a set $Q=\\{q_{1},q_{2},\ldots,q_{|Q|}\\}$ of $hospitals$;

  * (1.3)

set $Q$ denotes, at the same time, the set of $districts$ in which the State
is consequently partitioned [one district for one hospital]; in particular set
$Q$ is associated with a population distribution
$(a_{1},a_{2},\ldots,a_{|Q|})$ where component $a_{i}$ denotes the population
fraction of district $q_{i}$ (for $i=1,\ldots,|Q|$);

  * (1.4)

there is a set $R=\\{r_{1},r_{2},\ldots,r_{|R|}\\}$ of (types of) wards;

  * (1.5)

each hospital has got all (types of) wards; in particular, all wards of each
hospital are of the same level, we say that they are basic wards.

Let us assume that the evolution of this baseline scenario consists in the
evolution of hospitals which is defined as follows: the evolution of a
hospital consists in making wards excel, i.e., in transforming some of its
basic wards into excellent wards.

Such potential excellent wards are aimed at a ”new market” of patients [which
live in the State], formed by those patients who would be either transferred
outside the State or directly hospitalized outside the State, since doctors or
patients themselves would assume that State hospitals’ basic wards are not
suitable.

Then let us assume that the ”new market” is defined as follows:

  * (1.6)

the ”new market” is a set $P$ of $patients$ that, according to an estimation,
in a certain time frame should be hospitalized in a potential excellent ward
of type $r$ (for $r=1,\ldots,|R|$);

  * (1.7)

set $P$ is partitioned into groups $P_{1},P_{2},\ldots,P_{|R|}$, where $P_{r}$
is the group of patients that, according to an estimation, in a certain time
frame should be hospitalized in a potential excellent ward of type $r$ (for
$r=1,\ldots,|R|$); patients of group $P_{r}$ (for $r=1,\ldots,|R|$) are
distributed over districts according to population distribution
$(a_{1},a_{2},\ldots,a_{|Q|})$.

Then let us assume that the State has got a ”budget” to finance the evolution
of hospitals and that such a budget can take place by one of the following
financing systems:

\- local financing : the State splits the budget among the hospitals, each of
which is managing its own portion of the budget. Each hospital will pursue the
individual benefit: this benefit is defined as maximizing the number of
patients of the ”new market” which will be hospitalized in that hospital.

\- central financing : the State is the sole manager of the budget. The State
will pursue the whole community benefit: this benefit is defined as minimizing
the sum of costs to make the chosen wards (of the chosen hospitals) excel,
plus the sum of costs/discomforts that patients of the ”new market” should
incur to be hospitalized (either inside or outside the State).

Then let us try to study the evolution of this baseline scenario for such
financing systems.

##  3 Local financing

The study of local financing is inspired to reference [14]. For that, first we
report a part of reference [14] in which the authors study a scenario as a
Game Theory problem, then we study local financing as a similar Game Theory
problem.

For any reference about Game Theory let us refer to [8]. Let us recall that a
game is a scenario in which individuals (players) can choose respectively
certain behaviors (strategies), with the aim to maximize respectively their
benefits (payoffs), taking into account that their respective payoffs depend
on the strategies chosen by all players. The goal of Game Theory is to
predict, given a game and given certain assumptions on players’ rationality,
players’ behavior and therefore a game outcome. In particular a Nash
Equilibrium $-$ which is a basic concept which provides the most accepted
prediction of a game outcome $-$ of a game with two players is a pair of
strategies, one for each player, such that each player has no benefit in
independently changing his strategy, i.e., such that these two strategies are
mutually optimal responses.

###  3.1 Reference [14]

Reference [14] studies a scenario (see Case I of [14]) in which two hospitals,
say hospital I and hospital II, have the same ”market”, made up of three
groups of people: Sportspeople (22%), Women ( 46%), Children (32%). Each
hospital has funds to equip reception for only two groups of people. This
scenario corresponds to the following game (static with complete information):

:: Players: the set of players is {hospital I, hospital II};

:: Strategies: the set of strategies at disposal to each player is {SW, SB,
DB}, where SW is the strategy to equip reception for Sportspeople and Women,
SC for Sportspeople and Children, WC for Women and Children;

:: Payoffs: the payoffs that players get for each combination of strategies
are reported in the (bi) matrix in Figure 1; in particular, the authors assume
that if hospitals turn to the same group, then hospitals divide that group
into equal parts.

| $II:SW$ | $II:SC$ | $II:WC$
---|---|---|---
$I:SW$ | $33\%,33\%$ | $57\%,43\%$ | ${\bf 45\%},{\bf 55\%}$
$I:SC$ | $43\%,57\%$ | $34\%,34\%$ | $38\%,{\bf 62\%}$
$I:WC$ | ${\bf 55\%},{\bf 45\%}$ | ${\bf 62\%},38\%$ | $39\%,39\%$

Figure 1

For example, if I chooses SW and if II chooses SC, then: I and II divide the
share of Sportspeople (11% for each), I gets the share of Women (46%), II gets
the share of Children (32%); then, at the end, I gets 11% + 46% = 57%, II gets
11% + 32% = 43%.

This game has two Nash Equilibria, i.e., it has two predicted outcomes:

first predicted outcome: I choose SW, II chooses WC;

second predicted outcome: I choose WC, II chooses SW.

###  3.2 Local financing as a Game Theory problem

Let us assume that each hospital, with its portion of the budget, can make one
single ward excel. Then let us study local financing as the following game
(static with complete information).

Game $G$:

:: Players: the set of players is $\\{q_{1},q_{2},\ldots,q_{|Q|}\\}$;

:: Strategies: the set of strategies at disposal to each player is
$\\{[r_{1}],[r_{2}],\ldots,[r_{|R|}]\\}$, where $[r_{i}]$ is the strategy of
making ward of type $r_{i}$ excel (for $i=1,\ldots,|R|$);

:: Payoff: the payoffs that players get for each combination of strategies are
reported for the sole case $|Q|$ = 2 and $|R|=2$, in hospital $q_{1}$ and
hospital $q_{2}$, in the (bi) matrix in Figure 2; in particular, we assume
that if more hospitals choose strategy $[r_{i}]$ (for $i=1,\ldots,|R|$), that
is, they turn to the same group $P_{i}$ of the ”new market”, then these
hospitals divide the group $P_{i}$ according to the population distribution
$(a_{1},a_{2},\ldots,a_{|Q|})$.

| $q_{2}:[r_{1}]$ | $q_{2}:[r_{2}]$
---|---|---
$q_{1}:[r_{1}]$ | $|P_{1}|/a_{1},|P_{1}|/a_{2}$ | $|P_{1}|,|P_{2}|$
$q_{1}:[r_{2}]$ | $|P_{2}|,|P_{1}|$ | $|P_{2}|/a_{1},|P_{2}|/a_{2}$

Figure 2

Consider the following two numerical examples for the case $|Q|=2$ and
$|R|=2$.

Example 1.

Assume that: $|P_{1}|$ = 1000, $|P_{2}|$ = 400, ($a_{1}$, $a_{2}$) = (1/4,
3/4). Then one obtains the game in Figure 3.

| $q_{2}:[r_{1}]$ | $q_{2}:[r_{2}]$
---|---|---
$q_{1}:[r_{1}]$ | $250,{\bf 750}$ | ${\bf 1000},400$
$q_{1}:[r_{2}]$ | ${\bf 400},{\bf 1000}$ | $100,300$

Figure 3

This game has one Nash Equilibrium, that is one predicted outcome: $q_{1}$
chooses $[r_{1}]$, $q_{2}$ chooses $[r_{2}]$. $\Box$

Example 2.

Assume that: $|P_{1}|=1000$, $|P_{2}|=4$, ($a_{1}$,$a_{2}$) = (1/4, 3/4). Then
one obtains the game in Figure 4.

| $q_{2}:[r_{1}]$ | $q_{2}:[r_{2}]$
---|---|---
$q_{1}:[r_{1}]$ | ${\bf 250},{\bf 750}$ | ${\bf 1000},4$
$q_{1}:[r_{2}]$ | $4,{\bf 1000}$ | $1,3$

Figure 4

This game has one Nash Equilibrium, that is one predicted outcome: $q_{1}$
chooses $[r_{1}]$, $q_{2}$ chooses $[r_{1}]$. $\Box$

Example 1 and Example 2 have two different predicted outcomes: in Example 1,
hospitals diversify their choices, in Example 2 hospitals do not diversify
their choices. This seems to be linked to the fact that: in Example 1 the ”new
market” groups are of the same order of magnitude ($|P_{1}|=1000$,
$|P_{2}|=400$), while in the Example 2 the groups of ”new market”are not of
the same order of magnitude ($|P_{1}|=1000$, $|P_{2}|=4$).

In order to possibly draw general conclusions, we introduce the following
assumption, which will allow us to conclude that the tendency of hospitals is
to diversify their choices.

Assumption 1:

a) the components of population distribution ($a_{1}$, $a_{2}$) have the same
order of magnitude [as above in the two examples: ($a_{1}$, $a_{2}$) = (1/4,
3/4)];

b) the ”new market”. is formed by groups having the same order of magnitude
[as above in Example 1: $|P_{1}|$ = 1000, $|P_{2}|$ = 400].

Observation 1

Assumption 1-(a) seems to be plausible, by definition of (hospitals) district,
in the sense that (hospitals) districts are generally defined and saved in a
balanced way; Assumption 1-(b) seems to be plausible, by definition of (type
of) wards, in the sense that (types of) wards are generally defined and saved
in a balanced way. $\Box$

Assumption 1 implies [that is can be formalized by] the following
inequalities:

$|P_{k}|>min\\{|P_{i}|/a_{j}:i=1,\ldots,|R|;j=1,\ldots,|Q|\\}$ for
$k=1,\ldots,|R|$,

that for the case $|Q|=2$ and $|R|=2$ are:

$|P_{2}|>min\\{|P_{i}|/a_{j}:i=1,2;j=1,2\\}$;
$|P_{1}|>min\\{|P_{i}|/a_{j}:i=1,2;j=1,2\\}$,

that expresses the following condition: each group of the ”new market”,
corresponding to a certain type of ward, estimated on the whole State is more
numerous than the smaller subgroup of the ”new market”, corresponding to
another type of ward, estimated on any (hospital) district.

###### Proposition 1

If Assumption 1 holds true, then game $G$:

  * (i)

has no Nash Equilibrium in which all players choose the same strategy,

  * (ii)

has a Nash Equilibrium in which not all players choose the same strategy.

Proof. Since Assumption 1 holds true, the above inequalities hold true. The
proof is only for the case $|Q|=2$ and $|R|=2$ [cf. Table 5 below]. For the
case $|Q|=2$ and $|R|>2$, the proof can be carried out by a similar approach.
For the case $|Q|>2$ and $|R|>2$ the proof can be carried out by a similar
approach but with a greater level of detail, for that, let us omit to report
here for brevity and convenience.

| $q_{2}:[r_{1}]$ | $q_{2}:[r_{2}]$
---|---|---
$q_{1}:[r_{1}]$ | $|P_{1}|/a_{1},|P_{1}|/a_{2}$ | $|P_{1}|,|P_{2}|$
$q_{1}:[r_{2}]$ | $|P_{2}|,|P_{1}|$ | $|P_{2}|/a_{1},|P_{2}|/a_{2}$

Figure 5

Proof of (i). By contradiction let us assume that $G$ has a Nash Equilibrium
in which all players choose the same strategy, say without loss of generality
that ($[r_{1}]$, $[r_{1}]$) is a Nash Equilibrium. Then one has that
$|P_{1}|/a_{2}>|P_{2}|$ and that $|P_{1}|/a_{1}>|P_{2}|$. But this contradicts
at least one of the above inequalities.

Proof of (ii). Let us say that a pair $(x,y)$ of strategies, where $x$ is a
strategy of $q_{1}$ and $y$ is a strategy of $q_{2}$, is diversified if $x\neq
y$ and is non-diversified if $x=y$.

Then let us consider the following exhaustive cases.

Case 1: a pair of non-diversified strategies contains an optimal response.

Without loss of generality, let $([r_{1}],[r_{1}])$ be a pair of non-
diversified strategies containing an optimal response, and let strategy
$[r_{1}]$ be an optimal response of $q_{2}$ to strategy $[r_{1}]$ of $q_{1}$.
Then $|P_{1}|/a_{2}>|P_{2}|$. This implies that $|P_{1}|>|P_{2}|/a_{2}$ [in
fact, $|P_{1}|/a_{2}>|P_{2}|$ implies $|P_{1}|>|P_{2}|$, which in turn implies
that $|P_{1}|>|P_{2}|/a_{2}$] and that $|P_{1}|/a_{1}<|P_{2}|$, since
otherwise at least one of the above inequalities would be contradicted. These
two inequalities imply that $([r_{1}],[r_{2}])$ is a Nash Equilibrium of $G$.
Then, in Case 1, (ii) is proved.

Case 2: no pair of non-diversified strategies contains any optimal response.

Let us write $|P_{m}|=max\\{|P_{1}|,|P_{2}|\\}$, without loss of generality,
let $|P_{m}|=|P_{2}|$. Then, for each index $i\neq m$, an optimal response to
a player’s strategy $[r]$ is $[r_{m}]$ of the other player. On the other hand,
by the assumption of Case 2, one has that $([r_{m}],[r_{m}])$ does not contain
any optimal response. Let $[r_{j}]$ be an optimal response of $q_{1}$ to the
strategy $[r_{m}]$ of $q_{2}$ (without loss of generality): then, by the
above, one has that $j\neq m$ and that $([r_{j}],[r_{m}])$ is a Nash
Equilibrium of $G$. Then, in Case 2, (ii) is proved. $\Box$

Conclusion for local financing

In the local financing system: if Assumption 1 holds true, according to
Observation 1, then hospitals tend to diversify their excellences according to
Proposition 1.

##  4 Central financing

The study of central financing is inspired to reference [12]. For that, first
let us report a part of reference [12] in which the author studies a
Mathematical Programming problem, then let us study central financing as a
similar Mathematical Programming problem.

For any reference about Mathematical Programming, in particular about Integer
Linear Programming, let us refer to [3].

###  4.1 Reference [12]

Reference [12] studies the following well-known problem.

Plant Location: A certain number of plants must be activated, to be chosen
over a set of potential plants, which will have to supply a group of
customers. Activating the potential plant $j$ involves a cost $d_{j}$. It is
assumed that each facility has unlimited supply capacity. Each customer will
be connected to a single plant; in particular, connecting customer $i$ to
plant $j$ entails a cost $c_{ij}$. The problem is to choose which plants
should be activated in order to minimize the total costs for plants activation
and for customers connection.

This problem can be formulated in terms of Integer Linear Programming. For
this kind of problem $-$ which belongs to the class of NP-hard problems, see
e.g. [9], for an introduction to this kind of problems and for related topics
$-$ it becomes difficult to compute an optimal solution, from the point of
view of computation, when the problem involves a large number of variables
subject to integrality constraints. Then it is usual to determine, instead of
an optimal solution of the problem, a heuristic solution of the problem: that
is an admissible solution which is computed by a heuristic method, i.e., by a
quick and common sense method that provides a presumably good solution (which
is not necessarily an optimal solution). In particular, the heuristic solution
chosen in [12] for Plant Localization is that produced by a heuristic method
called the Greedy Algorithm.

###  4.2 Central financing as a Mathematical Programming problem

Let us study central financing as the following problem.

SGM: The State has a budget $B$ to make excel in some hospital some wards with
the aim to (possibly) hospitalize patients of $P$ (of the ”new market”).
Making excel in hospital $q$ ward $r$ entails a cost $C_{qr}$. It is assumed
that in each hospital each ward has sufficient reception capacity. Every
patient of $P$ will be hospitalized either in a made excel ward inside the
State or outside the State; in particular, hospitalizing patient $p$ in
hospital $q$ in a made excel ward $r$ entails a cost/discomfort $c_{pqr}$ (for
the patient), while hospitalizing patient $p$ outside the State entails a
cost/discomfort $c_{p,out}$ (for the patient). The problem [of the State] is
to choose, in each hospital, which wards should be made excel in order to
minimize the overall costs sustained by the State and by patients of $P$.

SGM can be formulated as a problem of Integer Linear Programming as follows:

:: Parameters

$B$ = budget available to the State to make excel some hospital wards.

For $p\in P$, for $q\in Q$, for $r\in R$:

$c_{pqr}$ = cost/discomfort for patient $p$ if $p$ is hospitalized in hospital
$q$ in the (made excel) ward $r$;

$c_{p,out}$ = cost/discomfort for patient $p$ if $p$ is hospitalized outside
the State;

$C_{qr}$ = cost to make excel in hospital $q$ the ward $r$.

:: Decision variables

$y_{qr}$ for $q\in Q$, for $r\in R$

$x_{pqr}$ for $p\in P$, for $q\in Q$, for $r\in R$

$x_{p,out}$ for $p\in P$

$y_{qr}=1$ if in hospital $q$ ward $r$ is made excel;

$y_{qr}=0$ otherwise;

$x_{pqr}=1$ if patient $p$ is hospitalized in hospital $q$ in the (made excel)
ward $r$;

$x_{pqr}=0$ otherwise;

$x_{p,out}=1$ if patient $p$ is hospitalized outside the State;

$x_{p,out}=0$ otherwise.

:: Model

min
$\sum_{q=1}^{|Q|}\sum_{r=1}^{|R|}C_{qr}y_{qr}+\sum_{p=1}^{|P|}\sum_{q=1}^{|Q|}\sum_{r=1}^{|R|}c_{pqr}x_{pqr}+\sum_{p=1}^{|P|}c_{p,out}$

subject to:

$x_{p,out}+\sum_{q=1}^{|Q|}\sum_{r=1}^{|R|}x_{pqr}=1$ for $p\in P$

(constrains to ensure that: every patient $p$ is hospitalized either in a
hospital $q$ in a ward made excel $r$ either outside of the State)

$x_{pqr}$ $\leq$ $y_{qr}$ for $p\in P$, for $q\in Q$, for $r\in R$

(constraints to ensure that: in hospital $q$ ward $r$ is made excel if at
least one patient $p$ is hospitalized therein)

$\sum_{q=1}^{|Q|}\sum_{r=1}^{|R|}C_{qr}y_{qr}\leq B$

(budget contraint)

$x_{pqr}\in\\{0,1\\}$ for $p\in P$, for $q\in Q$, for $r\in R$

$y_{qr}\in\\{0,1\\}$ for $q\in Q$, for $r\in R$

$x_{p,out}\in\\{0,1\\}$ for $p\in P$

(integrality constraints {0, 1})

Let us observe that, by the above, SGM may admit the following optimal
solution:

$x_{pqr}$ = 0 for $p\in P$, for $q\in Q$, for $r\in R$

$y_{qr}=0$ for $q\in Q$, for $r\in R$

$x_{p,out}=1$ for $p\in P$

that is an optimal solution such that in each hospital no ward is made excel
and all patients in $P$ are hospitalized outside the State.

This could happen for two occurrences:

(i) budget $B$ is insufficient to make in any hospital any ward excel;

(ii) once considered the costs to make in any hospital any wards excel and the
costs/discomforts (for patients in $P$) to hospitalize either inside or
outside the State, it is not advisable to make in any hospital any ward excel,
that is, all $P$ patients are hospitalized outside the State.

In order to avoid the above occurrences let us introduce the following
assumption.

Assumption 2: to avoid occurrence (i), we assume that $B\geq min\\{C_{qr}:q\in
Q,r\in R\\}$; to avoid occurrence (ii), we assume that the sum of terms
($c_{p,out}-c_{pqr}$) over indices $p,q,r$ is greater than the sum of terms
($C_{qr}$) over indices $q,r$: this assumption is consistent with the fact
that one presumably has $c_{p,out}\geq c_{pqr}$ (for a patient of $P$ it means
more cost/discomforts to be hospitalized outside the State rather than within
the State) and with the fact that patients of $P$ are numerous.

Let us report below a heuristic method, namely the Greedy Algorithm, to
compute a heuristic solution of SGM.

Let $Y=\\{y_{qr}:q\in Q,r\in R\\}$. A subset $T\subseteq Y$ is admissible if
the sum of values $(C_{qr})$ over indices $q$ and $r$ such that $y_{qr}\in T$
is less than or equal to $B$ [i.e. if the budget constraint is satisfied]. By
definition of the SGM problem, each admissible solution of the SGM problem is
associated with an admissible subset $T\subseteq Y$. In fact: on the one hand,
an admissible solution clearly identifies an admissible $T\subseteq Y$ subset;
on the other hand, once fixed an admissible subset $T\subseteq Y$, it remains
automatically defined a ”best possible solution” associated with $T$, which is
the solution in which patients in $P$ will be hospitalized to a destination
[either among those in $T$, or outside the State] which entails a lowest
cost/discomfort; in particular, for each admissible subset $T\subseteq Y$, we
denote as $Z(T)$ the value of the objective function for a ”best possible
solution” associated with $T$.

Thus SGM can be re-defined as follows:

SGM: Determine an admissible subset $T\subseteq Y$ such that $Z(T)$ is
minimum.

Let us report from [12], with appropriate modifications, the following Greedy
Algorithm to compute an admissible subset $T\subseteq Y$. The Greedy Algorithm
is a heuristic method, that is, a quick and common sense method provides a
presumably good solution (which is not necessarily an optimal solution), i.e.,
an admissible solution $T$ such that $Z(T)$ is presumably not far from the
optimal solution value of SGM.

Greedy Algorithm

Inizializzation: $i$ = 1, $T_{0}=\emptyset$, $Z(T0)=-\infty$.

begin of the $i$-th iteration

$y_{i}=argmin\\{Z(T_{i-1})\setminus\\{y\\}:y\in Y\ T_{i-1}\\}$.

If $Z(T_{i-1}\setminus\\{y_{i}\\})\geq Z(T_{i-1})$, then STOP: $T_{i-1}$ is
the greedy solution.

Otherwise set $T_{i}=T_{i-1}\setminus\\{yi\\}$.

If $T_{i}=Y$, then STOP: $T_{i}$ is the greedy solution.

Otherwise set $i=i+1$ and go to the $i$-th iteration.

end of the $i$-th iteration

Let us report from [12], with appropriate modifications, the following
observations on the Greedy Algorithm: once an element $y\in Y$ is inserted in
the (current) subset $T\subseteq Y$, then it is no longer removed; the element
to be inserted in the (current) subset $T\subseteq Y$ is chosen, at each step,
by looking just at the immediate advantage consisting in a decrease in the
value of the objective function.

Let us introduce the following assumption:

Assumption 3: we accept the greedy solution of SGM produced by the Greedy
algorithm as the solution to SGM even if it is not guaranteed to be an optimal
solution, according to the above and to the fact that SGM involves a large
number of variables subject to integrality constraints.

Then in order to possibly draw general conclusions [on the structure of the
greedy solution] let us also introduce the following assumptions.

Assumption 4: cost/discomfort $c_{pqr}$ depends only on the district in which
$p$ is located and on the distance from this district to district $q$ [i.e.,
it does not depend on $r$]; in particular, this cost is the same for every
person in a given district.

Assumption 5: $C_{qr}=K$, for each $q\in Q$, for each $r\in R$; that is, the
cost to make excel in hospital $q$ ward $r$ is a standard cost $K$ that does
not depend on $q$ and $r$.

Observation 2

Assumption 2 seems to be plausible since it is consistent with plausible
facts. Assumption 3 seems to be acceptable according to what is reported in
Subsection 4.1. Assumption 4 seems to be plausible at least in the general
case. Assumption 5 is expressed in an improbable manner, however it seems to
be plausible that all these costs have the same order of magnitude: then, in
this approximate context where the greedy solution is accepted as the solution
to the SGM problem, Assumption 5 seems to be plausible. $\Box$

Observation 3

It is possible to define two total orders, one for set $R$, one for set $Q$.

:: The first total order refers to set $R=\\{r_{1},r_{2},\ldots,r_{|R|}\\}$ of
(types of) wards according to the baseline scenario introduced above [cf.
(1.7)].

That is: we assume that $r_{1},r_{2},…,r_{|R|}$ are ordered so that
$|P_{1}|\geq|P_{2}|\geq\ldots\geq|P_{|R|}|$.

:: The second total order refers to set $Q=\\{q_{1},q_{2},\ldots,q_{|Q|}\\}$
of hospitals according to Assumptions 4-5.

That is, once fixed a (type of) ward $r$, we assume that
$q_{1},q_{2},\ldots,q_{|Q|}$ are ordered so that:

$q_{1}$ = hospital such that, wishing to make excel in exactly one hospital
the ward $r$, then the most convenient hospital is hospital $q_{1}$;

$q_{2}$ = hospital such that, wishing to make excel in exactly one hospital
the ward $r$, assuming that in hospital $q_{1}$ ward $r$ is already made
excel, then the most convenient hospital is hospital $q_{2}$;

more generally, for $k\geq 2$, we assume that:

$q_{k}$ = hospital such that, wishing to make excel in exactly one hospital
the ward $r$, assuming that in hospitals $q_{1},\ldots,q_{k-1}$ ward $r$ is
already made excel, then the most convenient hospital is hospital $q_{k}$.

This order is the same for all (types of) wards by Assumptions 4-5. $\Box$

Observation 4

The total orders $r_{1},r_{2},\ldots,r_{|R|}$ and $q_{1},q_{2},\ldots,q_{|Q|}$
defined in Observation 3 can be easily computed once the parameter values are
defined. $\Box$

Observation 5

The total order $q_{1},q_{2},\ldots,q_{|Q|}$ of hospitals (of Observation 3)
does not necessarily coincide with the total order of hospitals that would be
obtained on the basis of the population in the respective districts. For
example, in Abruzzo (Italy) the total order of hospitals on the basis of the
population of the respective districts is presumably: Pescara, Chieti,
L’Aquila, etc., while the total order $q_{1},q_{2},\ldots,q_{|Q|}$ of
hospitals (of Observation 3) is presumably: Pescara, L’Aquila, Chieti, etc.
This is since, once a ward $r$ is made excel at Pescara, then (from the point
of view of the minimization of costs) according to Assumption 4 it could be
agreed that the next ward $r$ is made excel at L’Aquila instead of at Chieti
although the population of (the district of) Chieti is greater than the
population (of the district) of L’Aquila. $\Box$

###### Proposition 2

If Assumptions 4-5 hold true, then the greedy solution of the SGM problem is
such that:

if $y_{qr}=1$, then $y_{q^{\prime}r^{\prime}}=1$ for every $q^{\prime}$ that
precedes $q$ in the total order $q_{1},q_{2},\ldots,q_{|Q|}$ defined in
Observation 3, and for every $r^{\prime}$ that precedes $r$ in the total order
$r_{1},r_{2},\ldots,r_{|R|}$ defined in Observation 3.

Proof. The proof derives directly by definition of the Greedy Algorithm and by
definition of the total orders in Observation 3. $\Box$

In other words, the greedy solution of the SGM problem is such that, with
reference to the total orders defined in Observation 3, if in hospital $q_{h}$
ward $r_{k}$ is made excel, then in hospital $q_{i}$, for $i=1,2,...,h$, (at
least) wards $r_{1},r_{2},\ldots,r_{k}$ are made excel.

For example, if $|Q|=5$ and if $|R|=7$, then the greedy solution of the SGM
problem is of the form:

in $q_{1}$: wards $r_{1},r_{2},r_{3},r_{4},r_{5}$ are made excel

in $q_{2}$: wards $r_{1},r_{2},r_{3},r_{4}$ are made excel

in $q_{3}$: wards $r_{1},r_{2}$ are made excel

in $q_{4}$: no ward is made excel

in $q_{5}$: no ward is made excel

Conclusion for central financing

In the central financing system: if Assumptions 2-5 hold true, according to
Observation 2, then the State tends to create poles of excellence according to
Proposition 2.

Acknowledgements. Please I would like to thank Prof. Fausto Di Vincenzo for
having explained to me his paper [7]. Then please I would like to witness that
just try to pray a lot and am not able to do anything without that - ad laudem
Domini.

## References

  * [1] C. Collins, A. Green, Decentralization and primary health care: some negative implications in developing countries, Int. J. Health Serv. 24(3) (1994) 459–475
  * [2] De Geyndt W., Does autonomy for public hospitals in developing countries increase performance? Evidence-based case studies, Soc. Sci. Med., 179 (2017) 74–80
  * [3] F. S. Hillier and G J. Lieberman, ”Introduction to Operations Research”, San Francisco (1967) Holden-Day.
  * [4] M. van Ineveld, J. van Wijngaarden, G. Scholten, Choosing cooperation over competition; hospital strategies in response to selective contracting, Int. J. Health Planning and Management 33(4) (2018) 1082-1092 doi: 10.1002/hpm.2583.
  * [5] H. Leli, O. Addulahi, and B. Tsofa, Public hospitals’ finance management systems, and accountability mechanisms in the context of decentralized health systems in low- and middle-income countries – A thematic review, AAS Open Res. 2:18 (2019) doi.org/10.12688/aasopenres.12962.1
  * [6] A. Maharani, D. Femina, G. Tampubolon, Decentralization in Indonesia: lessons from cost recovery rate of district hospitals, Health Policy Plan. 30(6) (2015) 718–727
  * [7] D. Mascia, F. Di Vincenzo, A. Cicchetti, Dynamic analysis of interhospital collaboration and competition: Empirical evidence from an Italian Regional health system, Health Policy 105 (2012) 273-281
  * [8] M. J. Osborne, A. Rubinstein, ”A Course in Game Theory”, (1994) MIT Press Books.
  * [9] Papadimitriou C.H., Steiglitz K., ”Combinatorial Optimization: algorithms and complexity”, (1998) Dover Publications, N.Y.
  * [10] B. Rechel, A. Maresso, A. Sagan, et al., Editors. Organization and financing of public health services in Europe: Country reports [Internet]. Copenhagen (Denmark): European Observatory on Health Systems and Policies; 2018. (Health Policy Series, No. 49.) Available from: https://www.ncbi.nlm.nih.gov/books/NBK507325/
  * [11] D. A. Rondinelli, J. R. Nellis, Assessing decentralization policies in developing countries: The case for cautious optimism. Development Policy Review, 4(1) (1986) 3–23
  * [12] A. Sassano, ”Modelli e algoritmi della Ricerca Operativa”, Ed. Franco Angeli (1999).
  * [13] B. Tsofa, S. Molyneux, L. Gilson, et al., How does decentralisation affect health sector planning and financial management? A case study of early effects of devolution in Kilifi County, Kenya, Int. J. Equity Health 16(1) (2017) 151
  * [14] W. W. Westhoff, C. F. Cohen, E. E. Cooper, J. Corvin, and R. J. McDermott, Cooperation or Competition: Does Game Theory Have Relevance for Public Health, American Journal of Health Education, 43:3 (2012) 175-183, DOI: 10.1080/19325037.2012.10872089