D. M. Weigt and L. A. Ca$\tilde{n}$izares et al.The SURROUND Mission

# The science behind SURROUND: a constellation of CubeSats around the Sun

D. M. Weigt†School of Cosmic Physics, Dublin Institute for Advanced Studies,
Dublin, Ireland ,Department of Computer Science, Aalto University, Aalto,
Finland  L. A. Ca$\tilde{\mathrm{n}}$izares†1,School of Physics, Trinity
College Dublin, Dublin , Ireland  S. A. Maloney1 S. A. Murray1
E. P. Carley1 P. T. Gallagher1 A. Macario-RojasSchool of Engineering, The
University of Manchester, Manchester, UK  N. Crisp4 and C. McGrath4
†these authors contributed equally to this work

###### Abstract

One of the greatest challenge facing current space weather monitoring
operations is forecasting the arrival of coronal mass ejections (CMEs) and
Solar Energetic Particles (SEPs) within their Earth-Sun propagation
timescales. Current campaigns mainly rely on extreme ultra-violet and white
light observations to create forecasts, missing out many potential events that
may be hazardous to Earth’s infrastructure undetectable at these wavelengths.
Here we introduce the SURROUND mission, a constellation of CubeSats each with
identical radio spectrometers, and the results of the initial Phase-0 study
for the concept. The main goal of SURROUND is to monitor and track solar radio
bursts (SRBs), widely utilised as a useful diagnostic for space weather
activity, and revolutionise current forecasting capabilities. The Phase-0
study concludes that SURROUND can achieve its mission objectives using 3 - 5
spacecraft using current technologies with feasible SEP and CME forecasting
potential: a first for heliospheric monitors.

##  1 The SURROUND Concept

Observing, analysing and forecasting potentially hazardous space weather
phenomena such as coronal mass ejections (CMEs: e.g., Webb & Howard (2012))
and solar flares (e.g., Fletcher et al. (2011)), have received increased
attention in recent years. For example, CMEs impacting the terrestrial space
environment can perturb and even compress the Earth’s geomagnetic field
pushing the radiation belts closer towards the planet whereas accelerated
charged particles from previous flaring activity have the potential to
damage/blind orbiting spacecraft. The potential impact of these events on our
technologies and infrastructure (e.g., GPS blackouts, overloaded power grids…)
is the main cause of this surge in research resulting many governments across
the world to recognise the effects of extreme space weather as a risk.***For
example, the UK government have included ‘severe space weather’ as a
‘Significant’ risk in their 2023 Risk register, an increase from ‘Moderate’ as
published in their 2020 report. Therefore many have invested in operations and
monitoring to track potentially hazardous events with the view that it is
imperative that forecasting and nowcasting such events are greatly improved,
to potentially avoid permanent damage to society.

Figure 1: Final descoped configuration of the SURROUND mission concept
composed of 5 CubeSats along the Sun-Earth ecliptic: three Lagrange spacecraft
(L1, L4 and L5) and two spacecraft deployed at orbits ahead and behind the
Earth with a simultaneous drift rate.

In an effort to improve monitoring and forecasting capabilities in space
weather observations, we propose the SURROUND concept mission, as part of the
European Space Agency (ESA) Open Space Innovation Platform (OSIP) Nanosats for
Spaceweather Campaign. In this proposal, a constellation of satellites in
ecliptic orbit around the Sun (see Figure 1; 3 Lagrange spacecraft and 2 on an
Earth-leading and lagging orbits similar to the Solar Terrestrial Relations
Observatory [STEREO; Eyles et al. (2009)]) are used to provide co-temporal
measurements that, when analysed using multilateration techniques (e.g.,
Bancroft (1985); Smith & Abel (1987)), will enable localisation and tracking
of solar events. Through consideration of appropriate multilateration
techniques and use cases, this paper defines suitable measurement requirements
and summarises the key findings of the Phase-0 study. The Phase-0 is to
effectively assess the feasibility of the SURROUND mission concept. Having
multiple nanosatellites dedicated to operational monitoring of the
heliosphere, for the first time, allows us to improve current forecasting
capabilities of these hazardous space weather phenomenon. This paper outlines
the key scientific and mission requirements of the proposed SURROUND mission
concept.

Current space-based solar observatories [e.g., Solar Orbiter (SolO; Müller et
al. (2013)), Parker Solar Probe (PSP; Fox et al. (2016))] and operational
forecasting centres [e.g., STEREO, Geostationary Operational Environmental
Satellites (GOES; Goodman et al. (2013))] monitor and observe the solar
surface and lower atmosphere over multiple wavelengths (i.e., white-light,
extreme ultraviolet (EUV), X-rays); very few solely image the heliosphere.
Therefore, although previous and ongoing campaigns have provided well
documented catalogues of various space weather hazards (CME and flare onsets,
solar wind evolution…), tracking the progress of eruptive events and thus
forecasting the probabilistic nature of whether these events will impact the
Earth’s space environment still remains one of the greatest challenges in the
field. Furthermore many of these phenomena are unobservable in EUV and white
light wavelengths directly. For example, many CMEs can appear very faint in
the lower corona and travel slower than typical events (i.e., ‘stealth CMEs’;
e.g., O’Kane et al. (2019) and references therein). These may be overlooked,
hindering possible forecasts. In addition, flare-associated Solar Energetic
Particles (SEPs) are by definition not accompanied by a discernable CME in
coronagraph observations and do not emit and/or scatter radiation at EUV and
white light wavelengths. However in both cases, associated electrons are
observable from in situ and solar radio measurements. This is particularly
useful for SEP events which may be hazardous to spacecraft and communications.
The lighter, less harmful electrons travel faster than the potentially
hazardous heavier protons, providing a useful diagnostic to forecast such
space weather events. These accelerated electrons appear as distinct features
in dynamic spectra produced from radio spectrometry: Solar Radio Bursts
(SRBs).

SURROUND will focus on tracking Type-II (associated with particle acceleration
generated in the shock fronts of CMEs as they propagate through the
interplanetary medium (Cane et al., 1982; Webb & Howard, 2012)) and Type-III
(SEPs linked to electron beam acceleration travelling on open heliospheric
magnetic field lines, generated from solar flare events (Reid & Ratcliffe,
2014)) spectral types, over a frequency range of 0.0225 (i.e., low frequency
solar radio emissions). Both can be identified from dynamic spectra produced
from radio spectrometry over this frequency range (see Figure 2). Space
weather monitoring facilities use Type-II and Type-III SRBs as diagnostics for
incoming CMEs and accelerated SEPs respectively. With this, the goal of space
weather operations is to provide accurate forecasting/nowcasting information
within the propagation times of these space weather events, using the radio
emissions from accelerated electrons as our diagnostics: within 12 hours to
$\sim$ 9 days for CMEs (Webb & Howard, 2012), and $\sim$ 2128min for the
lighter electrons from SEP events with the heavier protons following $\sim$ a
few hours behind (e.g., Reames (2013); Reid & Ratcliffe (2014)). With multiple
viewpoints to track any SRBs travelling towards Earth, SURROUND will be the
flagship for tracking and monitoring space weather.

Figure 2: A (left) cartoon showing various eruptive events and (right) example
spacecraft data (Wind/Waves data from Gopalswamy et al. (2019)) of a radio
dynamic spectra highlighting different spectral classes of SRBs. Type-II
bursts are associated with the shock-front of a CME while Type-IIIs are linked
with the acceleration of SEPs. Solar eruption illustration is adapted from
ESA/A. Baker, CC BY-SA 3.0 IGO. Image source.

##  2 Tracking Solar Radio Bursts (SRBs)

The SURROUND mission is designed to track Type-II and -III SRBs using
multilateration techniques. Such methods are designed for multiple receivers
to provide information on the location and approximate size of the source of
interest.

###  2.1 Modelling the best case scenario

The easiest multilateration technique to implement is the Time-of-Arrival
(TOA) method which is reliant on temporal knowledge the emission of the source
signal. The distance from the source can be determined from the time
difference between the time of arrival ($t_{\mathrm{arrival}}$) and the time
of emission ($t_{\mathrm{emission}}$), assuming a constant propagation
velocity $c$ (for radio emissions, this is the speed of light in a vacuum: $c$
$\approx\ 3.00\times 10^{8}$ ms-1):

$d=c(t_{\mathrm{arrival}}-t_{\mathrm{emission}})$ (1)

For a single receiver this corresponds to a circle with radius length $d$ of
which the circumference defines all possible source locations detected by the
particular receiver, as shown in Figure 3. The location of the source can be
solved by use of parametric equations. If there are at least three receivers
available, 2D multilateration can be performed (similarly four receivers is
needed for the three-dimensional case). This is given by the following
equation of the circle

$d_{r}^{2}=(x_{r}-x)^{2}+(y_{r}-y)^{2}$ (2)

where $d_{r}$ and ($x_{r},y_{r}$) are the distance from the source and
positional 2D coordinates of each individual receiver and ($x,y$) are the
source coordinates.

For the case of SURROUND, the spatial and temporal information of an emitted
SRB is unknown and therefore this technique is not practical from an
operations perspective. However, TOA can be used to describe the “best case”
scenario for tracking these bursts with very close overall performance to more
complex methods (e.g., Kaune (2012)). Here we use a fully geometrical TOA
method as an ideal case to compare with a Bayesian algorithm where the
probabilistic position of the SRB or posterior distribution, $P(x,v|\Delta
t)$, is calculated given that there is an observing difference in time $\Delta
t$ between source and reciever without any given knowledge of the signal wave
propagation speed (e.g., Salvatier et al. (2016)). This is described by a
version of Bayes’ Theorem considering three parameters:

$P(x,v|\Delta t)=\frac{P(\Delta t|x,v)P(x|v)P(v)}{P(\Delta t)}$ (3)

where $x$ and $v$ are the source position and wave propagation velocity. The
data likelihood, $P(\Delta t|x,v)$, is the physics model used in the algorithm
- for the case of SURROUND we modelled the source-receiver distance from each
satellite with added noise to simulate possible propagation effects. The prior
distributions for the given parameters ($x,v$ and $\Delta t$) are given by
$P(x|v),P(v)$ and $P(\Delta t)$ where $P(x|v)$ denotes the prior distribution
of the source position given the propagation velocity of the source (or SRB).

Figure 3: Schematic of how both the (top-left) TOA and (top-right) TDOA
techniques determine the location of a given source. Source location is
determined from the intersection of circumferences and hyperbola pairs for TOA
and TDOA respectively. Bottom panel shows a simplified cartoon of the
goniopolarimetry (GP) method with antenna beam pattern and resultant
reconstructed wave power back-propagated to source (red).

###  2.2 Multilateration and direction finding with SURROUND

The time-difference-of-arrival (TDOA) multilateration technique is more
complex than TOA as it relies on the difference between the times of arrival
of the signal at each receiver (or satellite) to localise the source.
Therefore the TDOA method is independent of the time of emission and more
applicable to the scenarios SURROUND will encounter. As a result, we obtain a
difference in distance between the receivers to the source:

$\Delta
d_{i,j}=c(t_{i}-t_{j})=\sqrt{\left(x_{j}-x\right)^{2}+\left(y_{j}-y\right)^{2}}-\sqrt{\left(x_{i}-x\right)^{2}+\left(y_{i}-y\right)^{2}}$
(4)

where $t_{i,j}$ is the time of arrival to receivers $i$ and $j$;
($x_{i,j},y_{i,j}$) are the known positions of receivers $i$ and $j$ and
($x,y$) is the hyperbolic solution for the position of the source. From these
solutions, TDOA uses multiple satellites to isolate an area where the
likelihood of the source location is high (as shown in Figure 3). Therefore
increasing the number of satellites and deploying them in a calculated,
strategic position around the source emission will decrease the error in
source location (i.e., the area of intersection between pairs of hyperbola
decreases), given the separation angle is large enough. For multiple radio
sources (e.g. along a CME shock front) the calculated TDOA will be different
and therefore in a different position in the inner heliosphere. To disentangle
these sources, we can trace along a distinctive feature of the SRB on the
dynamic spectra by fitting a line profile (e.g., leading edge for Type-III
bursts) to find the TOA at each satellite and use multilateration to determine
the position across multiple frequency channels.

The main assumption used in both of TDOA and TOA is that the measurements are
based on the shortest target-receiver path the signal can take, or the Line of
Sight. The signal will be very weak or unobserved if this path is blocked,
leading to erroneous results in the location or possibly very large
uncertainties.

SURROUND will also be capable to fully implement the goniopolarimetry (GP)
method (numerical solutions found in Cecconi & Zarka (2005)). GP functions by
determining the propagation vector of detected electromagnetic waves emitted
from the source using three calibrated and carefully positioned antennae (or
two with one acting as a spinning dipole). Each antenna (or dipole antenna)
measures the polarisation and flux of the detected radio emissions which are
then used to reconstruct the resulting wave, and back propagate to the source
within a systematic uncertainty (dependent on parameters of the radio antenna;
see simple configuration in Figure 3). For the case of SURROUND, using GP with
multiple CubeSats observing radio signals from the same source, will result in
each individual wave vector being back propagated to the same position. Where
these wave vectors intersect will provide the location of the radio source
when observed at the same frequency across all spacecraft (required to perform
GP). Full modelling and more rigorous analysis of the GP method will be
carried out in the next mission phase.

This method has been implemented successfully from using a single spacecraft
(e.g., analysis of Saturn’s kilometric radiation from Cassini (Cecconi et al.,
2009)) and multiple spacecraft (e.g., tracking and triangulating a Type-III
burst using Wind and Ulysses (Reiner et al., 1998)); both methods treating the
source as point-like. For the case of SRBs however, due to scattering in the
interplanetary medium, the SRBs will appear spatially extended (Cecconi, 2007)
and therefore the original GP treatment from Cecconi & Zarka (2005) need to
account for this. This has been implemented in previous studies tracking Type-
II (e.g., Martínez Oliveros et al. (2012)) and -III SRBs (e.g., Krupar et al.
(2012)). Therefore the expertise on calibrating the antenna and analytically
solving the equations dealing with the correlations between all three antennae
for an extended source is all readily available.

In the case of SURROUND, GP capable antennas would be beneficial to the
mission as it will allow SRB localisation and tracking to be performed at
small separation angles when TOA/TDOA is very inaccurate and/or fails.
Additional information of the radio bursts (e.g., polarisation) detected from
a 3-antenna setup will be useful for further science cases (see Section 3).
Given the low directivity of the Type-II and -III emissions we are interested
in, the errors produced from multilateration are minimised and the Poynting
vectors for GP (e.g., resulting wave vectors) are not parallel when the
satellites are at varied distances from the Earth with a reasonable separation
angle. Therefore these methods take advantage of the low directivity to
improve their performance.

As solving the GP equations is more complex than the TOA/TDOA methods, this is
computationally more expensive. Therefore for providing forecasts within
30-min (see Section 3) the multilateration methods will be preferable.
However, the same antenna configuration can be applied to both sets of methods
reducing the complexity of our payload. We do note that these methods for
tracking SRBs initially assume that the radio signal produced from the source
is isotropic and is unaffected by inharmonious scattering along the signal
path, and that any errors will be dominated by the uncertainty in source
location. Although, as noted in previous studies (e.g., Bonnin et al. (2008)),
the effective beam of the source (electron beam with scattering effects) is
significantly anisotropic and will affect our source location. Such effects
will be explored and analysed in detail in the later stages of the mission
after Phase-0 (i.e., Phase-A).

##  3 Mission Requirements

The science goals of SURROUND can be summarised in the following statement:
“The SURROUND mission will detect, localise and track solar radio bursts in 3D
using a constellation of radio spectrometers and to forecast the arrival of
the radio bursts’ drivers at Earth (or Earth-like distances).”

The aim of this section is to synthesise the above mission statement and
briefly explain the drivers and requirements motivating the SURROUND concept.
Here we start by explaining the Use Cases of the mission; preliminary results
and outcomes for the use of multilateration techniques to track SRBs and then
conclude by discussing the Measurement requirements needed to fulfil the
science goals and drive the design of the mission and spacecraft.

###  3.1 Use Cases

The SURROUND mission will focus on space weather operations and therefore the
primary use case of the concept mission is the detection, tracking and
localisation of SRBs. This will provide us with a novel and more accurate way
of forecasting the arrival of these events at the Earth. As discussed
throughout this Phase-0 study, SURROUND will primarily focus on two types of
SRBs: (i) Type-II and (ii) Type-III bursts. SURROUND’s main Use Cases (UCs)
focus on detecting and tracking SRBs (Table 1: UCs 1 and 2); localise and
determine the location of SRBs (UCs 3 and 4); providing nowcasting results of
CMEs and SEP electrons (UCs 5 and 6) and forecast the arrival of space weather
drivers within their propagation timescales (UCs 7-9). These Use Cases are
summarised in Table 1 and will drive the Measurement Requirements to determine
the specifications of the payload needed (e.g., length of antenna, frequency
range, sensitivity…).

Table 1: Summary of SURROUND’s Use Cases | Use Case
---
(UC)
Description |
UC1 | Detect and track Type-II radio bursts - fast CMEs shocks accelerated electron beams |
UC2 | Detect and track Type-III radio bursts - flare accelerated electron beams |
UC3 | Localise in 3D Type-II radio bursts |
UC4 | Localise in 3D Type-III radio bursts |
UC5 | Nowcast CMEs associated with Type-IIs |
UC6 | Nowcast SEP electrons associated with Type-IIIs |
UC7 |  | Forecast arrival of CMEs associated with Type-II at Earth* (propagation time
---
scale $\sim$12 hours)
UC8 |  | Forecast arrival of SEP electrons associated with Type-III at Earth* (propagation
---
time scale $\sim$10 minutes)
UC9 |  | Forecast the potential arrival of SEP protons associated with Type-III
---
(propagation time scale $\sim$2 hours)
UC10† | Scientific Analysis |
∗ or at Earth like distances
† SURROUND’s secondary Use Case

The secondary Use Case proposed for SURROUND is scientific analysis of
obtained mission data. While the primary objectives focus on providing
operational information for space weather monitoring and forecasting, the
gathered data will be beneficial for many in the solar and planetary science
community. For example, a minimum of three different viewpoints of the solar
radio emissions, all providing simultaneous measurements, allows the Sun’s
global behaviour to be observed in finer detail when compared to current
observatories. The multilateration techniques can also incorporate other
solar/heliospheric monitors further improving the accuracy of SURROUND’s
localisation and tracking measurements (e.g., further reducing the area of
intersection). Utilising these combined results, or solely SURROUND
measurements, will improve the potential now/fore-casting capabilities of the
mission itself.

###  3.2 Measurement Requirements

Using the Use Cases as a foundation (Table 1), we identify the main
requirements essential for the SURROUND mission to achieve its science
objectives. These are outlined in Table 2 with the corresponding numbered Use
Case(s) that influenced the noted requirement and are essential to the next
stages of the mission analysis. Although the main focus of this paper is more
the science aspect of the SURROUND mission, these stages, together with the
Mission Requirements, will determine the final outcomes of this Phase-0 Study
(as outlined in Section 4).

Table 2: Summary of SURROUND’s Measurement Requirements | MR*
---
Description | Key Driver |  | Use
---
Cases
MR1 |  | Detect radio waves with spectral
---
flux density of greater than
10e-22-2-1
| Detection of Type-II and
---
Type-III radio bursts
| UC1,
---
UC2
MR2 |  | Detect radio waves with in frequency
---
range of 0.0225
| Track radio burst from close to
---
solar ($\sim$2 $R_{\odot}$ surface
to 1
| UC1,
---
UC2
MR3 |  | Scan the frequency range of
---
0.0225
at least every 10
| Achieve accurate 3D localisation
---
using TOA/TDOA methods
| UC1,
---
UC2
MR4 |  | Minimum of 3 observation
---
locations (e.g., $L_{1}$,$L_{4}$, $L_{5}$) with
separation of at least $None$
| TOA/TDOA methods require at
---
least 3 different viewpoints (GP
has minimum of 2)
| UC3,
---
UC4
MR5 |  | Obtain data from spacecraft at least
---
every 6 hours
| Required time to process, generate
---
and issue nowcasts for Type-IIs
UC5 |
MR6 |  | Obtain data from spacecraft at least
---
every 15 minutes
| Required time to process, generate
---
and issue nowcasts for Type-IIIs
UC6 |
MR7 |  | Obtain data from spacecraft at least
---
every 3 hours
| Required time to process, generate
---
and issue forecasts for CMEs
associated with Type-IIs
UC7 |
MR8 |  | Obtain data from spacecraft at least
---
every 10 minutes
| Required time to process, generate
---
and issue forecasts for SEP electrons
associated with Type-IIIs
UC8 |
MR9 |  | Obtain data from spacecraft at least
---
every hour
| Required time to process, generate
---
and issue forecasts for potential SEP
protons associated with Type-IIIs
UC9 |
*MR = Measurement Requirement

The mission requirements can be grouped into four categories which we define
and briefly expand upon here:

  1. 1.

Frequency range and sensitivity (MR1, MR2): The frequency range of the
antennas (needed for all multilateration methods) is calculated from the
distances SURROUND will be able to detect, track, localise and forecast SRBs
using plasma density models (Leblanc et al., 1998; Parker, 1958; Saito et al.,
1977). These density models suggest that SURROUND will be capable of tracking
bursts from 2 $R_{\odot}$ (or 2 $R_{\mathrm{Sun}}$: near the solar surface) to
215 $R_{\odot}$ (Earth’s orbit) corresponding to a frequency range of 0.225.
The sensitivity of SURROUND’s antennas should be such that radio waves with a
spectra flux density exceeding 10e-22-2-1 should be detectable. Both frequency
and sensitivity specifications are feasible for SURROUND and have been
successfully implemented on previous missions (i.e., plasma and radio
instrument on SolO (Maksimovic et al., 2020)). The capability of tracking SRBs
to 215 $R_{\odot}$ will enable us to provide multiple data of time evolution
of the electrons as they propagate through the inner heliosphere for
scientific analysis (secondary Use Case) to obtain useful diagnostics of the
source plasma (i.e. temperature, density profile).

  2. 2.

Time resolution (MR2, MR3): The time cadence of the spectrometers used in
SURROUND is essential in order to obtain the best quality data at an optimised
resolution. Using multilateration techniques also introduces a time
resolution-dependent error on the location of the radio source with time
cadence (the higher the cadence, the smaller the error on location). However
the higher time cadence leads to a larger volume of data that needs to be
processed and transmitted back to ground and therefore also needs to be
considered when addressing the data resolution. For a SURROUND mission
consisting of 5 spacecraft in the ecliptic plane, a 10 cadence is desirable
for all spectrometers. These data files can be transmitted to the ground
receiver during days of high solar activity and increased likelihood of solar
flaring. Quieter times of less priority will be measured using a 60s cadence
(i.e., lower time resolution). Similar results were also obtained for a
4-spacecraft mission in the event of one CubeSat being lost.

  3. 3.

Spacecraft orbits and position (MR4): In order to achieve the science
objectives and maintain sufficient accuracy for tracking and localising SRBs,
SURROUND will require at least 3 - 5 satellites. As demonstrated by the
initial spacecraft configuration shown in Figure 1, each satellite will be
placed in strategic orbits to maintain sufficient angular separation to
perform TOA/TDOA and optimised viewing of the Earth-facing of the Sun at all
times.

  4. 4.

Frequency of downlinks (MR5 - MR9): The frequency of downlinks depends on the
type of space weather phenomena being monitored depending on their Sun-Earth
travel time. For example, radio emissions produced from Type-III SRBs will
take $\sim$ 8min to reach the SURROUND spacecraft. The associated electrons,
travelling at $\sim$ 0.3c, will reach Earth in $\sim$ 30min. Current
capabilities can allow us to assume data processing and transmission will take
5min (e.g., $\sim$ 15-min buffer to generate forecast for electrons).
Therefore data acquisition needs to be at a higher frequency (e.g., 10-min).
The frequency has to be optimised to allow the burst to travel far enough to
derive propagation direction but fast enough to enable the data to be
processed and issue a forecast well below the typical propagation time (see
Table 2). The size of the data products to be communicated to the ground
receiver will also affect the downlink time. For SURROUND, we estimate each
antenna will have a daily data rate of $\sim$ 120150MB (depending on time
cadence) in broad agreement with the radio and plasma waves (RPW) instrument
onboard SolO (assuming a 256 spectral channel spectrometer with 12-bit
resolution and 35% margin accounting for overheads and general housekeeping of
the data).

##  4 Outcomes of SURROUND Phase-0 study

Based on the results from the work presented here and various discussions and
feasibility studies we can draw our final outcomes of this Phase-0 study
including, but not limited to:

  * •

SURROUND can achieve its mission objectives using either a 3, 4 or
5-spacecraft (in order of highest uncertainty of SRB localisation).

  * •

Deployment to $L_{4}$ and $L_{5}$ are the most challenging and requires most
propulsion. $L_{1}$ spacecraft, closest to Earth, requires orbit maintenance
over mission lifetime.

  * •

Tracking and localisation of various SRBs can be carried out using
multilateration and GP techniques, using the same antenna configuration.

  * •

With current technological and ground communication constraints (need to
retrieve from 1), data can be preferentially received at high cadence from
$L_{1}$ (every 30-mins to 1-hr) and at longer regular intervals from the
remainder of the constellation (at lower cadence).

  * •

Forecasting potential SEP and incoming CMEs is feasible (time cadence
dependent) given current technologies.

The results of this Phase-0 study suggest that the SURROUND is feasible and
providing potential solutions for the more challenging obstacles (e.g.,
spacecraft deployment/maintenance, data downlink timings for forecasts). There
is no other space weather monitoring mission (currently ongoing or in advanced
planning stages) like SURROUND, which will revolutionise how we track space
weather and how we can utilise CubeSats for deep space science.

## Acknowledgements

The SURROUND phase-0 study is funded by European Union’s Horizon 2020 research
and innovation programme under Grant agreement No. 952439 and project number
AO 2-1927/22/NL/GLC/ov as part of the ESA OSIP Nanosats for Spaceweather
Campaign

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# Learning to agree over large state spaces111I thank Hannu Vartiainen for
invaluable support and supervision. I also thank Hannu Salonen, Mark
Voorneveld, Klaus Kultti, Aviad Heifetz, Mats Godenhielm and two anonymous
reviewers for helpful comments that greatly improved the paper, and the
audiences of the 41st Annual Meeting of the Finnish Economic Association, the
15th European Meeting on Game Theory, Bayes By the Sea Conference 2019 and the
Lisbon Meetings in Game Theory and Applications 2019. Financial support from
the OP Group Research Foundation, the Yrjö Jahnsson Foundation, the Finnish
Cultural Foundation and the University of Helsinki is gratefully acknowledged.
All errors are mine.

Michele Crescenzi
<EMAIL_ADDRESS>University of Helsinki and Helsinki Graduate
School of Economics, Finland

(January 2022
This version is published in the Journal of Mathematical Economics (2022),
DOI: https://doi.org/10.1016/j.jmateco.2022.102654)

###### Abstract

We study how a consensus emerges in a finite population of like-minded
individuals who are asymmetrically informed about the realization of the true
state of the world. Agents observe a private signal about the state and then
start exchanging messages. Generalizing the classical model of rational
dialogues of Geanakoplos and Polemarchakis (1982) and its subsequent
extensions, we dispense with the standard assumption that the state space is a
probability space and we do not put any bound on the cardinality of the state
space itself or the information partitions. We show that a class of rational
dialogues can be found that always lead to consensus provided that three main
conditions are met. First, everybody must be able to send messages to
everybody else, either directly or indirectly. Second, communication must be
reciprocal. Finally, agents need to have the opportunity to engage in
dialogues of transfinite length.

KEYWORDS: agreement theorem, common knowledge, consensus, learning, rational
dialogue

JEL CLASSIFICATION: C70, D82, D83

##  1 Introduction

This is a paper on common knowledge acquisition and consensus. We are
interested in the following problem. Consider a situation of incomplete
information with finitely many agents. Each agent privately observes a
partitional signal, whose realization depends on the true state of the world.
Afterwards, agents can exchange messages, whose content depends on the private
information each agent has. What are the conditions under which everyone
eventually sends the same message? In other words, what are the conditions
under which a consensus eventually emerges? And when a consensus is attained,
is it common knowledge that everyone agrees?

Originally investigated by Geanakoplos and Polemarchakis (1982), the problem
of rational dialogues leading to consensus has been studied in two distinct,
yet not mutually exclusive, settings. First, in probabilistic environments
like Geanakoplos and Polemarchakis (1982), the state space is a probability
space and the messages exchanged are posterior probabilities about a fixed
event. Second, in more general environments like Bacharach (1985) or Krasucki
(1996), the state space is not necessarily a probability space, and the
messages exchanged are values of some abstract function. In addition, agents’
information partitions are assumed to be finite or, as in Parikh (1992),
countably infinite.

Our goal is to study rational dialogues and consensus in a strictly more
general setting in which the state space is just an arbitrary set, the
messages exchanged are values of some abstract function, there are no bounds
on the cardinality of the state space itself or the information partitions,
and the length of a dialogue can be any ordinal number. Differently put, we
study the emergence of consensus starting from a minimalist setting where we
have a non-empty set as the state space, an information partition for each
agent, and a well-defined function whose values are communicated in an
arbitrarily long dialogue. We assume that agents communicate truthfully, do
not tell lies and are not driven by strategic motives. The reason for
exploring such a generalization is that, strictly speaking, the concepts of
knowledge, common knowledge and consensus do not call for a probabilistic
structure of the state space. Nor do they require a particular bound on the
cardinality of the state space itself or the information partitions. Of
course, limitations of this kind can be needed in particular cases, for
example when the dialogue is about posterior probabilities. Nonetheless, such
limitations are not necessary to formulate the general question of when a
dialogue between rational agents leads to consensus.

Our contributions are the following. First, we provide a general framework for
the study of rational dialogues that induce a consensus. This framework
consists of finitely many agents, an arbitrarily large state space, an
abstract function whose value are communicated in dialogues of transfinite
length, and private communication. Previous results in the literature can be
obtained as particular cases of our model. Second, we give sufficient
conditions for the emergence of consensus. More specifically, provided that
the function whose values are communicated is sufficiently regular, a
consensus emerges if the communication protocol has two properties. Everyone
should be able to talk to everyone else during a dialogue, either directly or
indirectly. And communication should be reciprocal: if agent $i$ sends her
message to $j$ at some point during a dialogue, then $j$ has to send a message
back to $i$. These two conditions on the protocol are a natural generalization
of those already provided by Krasucki (1996) for finite models. Finally, we
write the model, and prove the main results, in such a way to exploit the
lattice structure of the set of all partitions of the state space. The
standard approach in the literature is to take a communication protocol as a
primitive. Any such protocol induces a graph over the set of agents, and
conditions ensuring consensus are found by studying the properties of this
graph. Our approach takes the opposite route. The primitive object is a graph
which describes who talks to whom during one round of communication. This
graph induces a self-function in the set of profiles of information
partitions: intuitively, it maps the information that agents have at the
beginning of the communication round to the refined information they have
after having talked. By iterating this function, we can generate a sequence of
profiles of partitions that captures all the information generated during a
dialogue. In so doing, we can work directly in the complete lattice of
information partitions and prove convergence to consensus with relative ease.

As we mentioned, we allow dialogues to have transfinite length. The reason is
that, when the state space is infinite, agents might have to exchange
infinitely many messages before reaching a commonly known consensus. This
point is illustrated in the example in Section 2. Now, the question arises:
How to make sense of a dialogue whose length is indexed by some infinite
ordinal? From a mathematical point of view, the concept of an infinitely long
dialogue is fully legitimate and well-defined. Moreover, such dialogues have
already been studied in Parikh (1992) and Aumann and Hart (2003). The first
paper explores common knowledge acquisition and consensus in a framework akin
to ours, but with only two agents and a countably infinite state space. The
other paper studies games played at the end of an infinitely long dialogue
whereas we do not explore game-theoretic situations. Nonetheless, we are aware
that the concept of a transfinite dialogue may sound fanciful and we hope the
following points will partly overcome skepticism. First, we emphasize that we
do not aim for practicality in this paper. Rather, our goal is to explore the
formal structure of dialogues that, at least in principle, are conducive to a
commonly known consensus. Second, dialogues of any length can be seen as
interactive inferential processes, in which everyone draws conclusions from
what everyone says or does, everyone draws conclusions from everyone’s
conclusions drawn from what everyone says or does, and so on ad infinitum.
Under this interpretation, we can phrase the key question of our paper as
follows: How long should an interactive inferential process be in order for
rational people to attain a commonly known consensus? An analogous question
has been studied in game-theoretical contexts. Specifically, Lipman (1994)
shows that in some games it is possible to attain common knowledge of
rationality only through a transfinite process of elimination of never-best
replies. Finally, we do not claim that transfinite dialogues are necessary in
every possible situation covered by our model. As we show in the example in
Section 2, transfinite dialogues become necessary when the message function
through which agents communicate has a low expressive power and, consequently,
not much information can be learned at each stage of the dialogue.

The rest of the paper is organized as follows. The next section contains an
example of a rational dialogue in which it takes infinitely many stages to
reach a commonly known consensus. The model is presented in Section 3 and the
results are illustrated in Section 4. Finally, in Section 5 we discuss the
related literature and some of the assumptions we make in our analysis.

##  2 Example

Consider a situation of incomplete information. There are three persons: Ann
($A$), Bob ($B$) and Carol ($C$). The set of possible states of the world is
the set of natural numbers $\mathbb{N}$. The true state cannot be observed
directly. Each person has access to a partitional signal whose realizations
are privately observed. The resulting information partitions are as follows.

$\displaystyle g^{0}_{A}$
$\displaystyle=\left\\{\\{1,2,7\\},\\{3,4\\},\\{5\\}\right\\}\cup\left\\{\\{6+4k_{1},9+4k_{1}\\}:k_{1}\geq
0\right\\}\cup\left\\{\\{8+4k_{2},11+4k_{2}\\}:k_{2}\geq 0\right\\}$
$\displaystyle g^{0}_{B}$
$\displaystyle=\left\\{\\{1+4k_{1},2+4k_{1}\\}:k_{1}\geq
0\right\\}\cup\left\\{\\{3+4k_{2},4+4k_{2}\\}:k_{2}\geq 0\right\\}$
$\displaystyle g^{0}_{C}$ $\displaystyle=\left\\{\\{1+4k_{1}:k_{1}\geq
0\\},\\{2+4k_{2}:k_{2}\geq 0\\},\\{3+4k_{3}:k_{3}\geq
0\\},\\{4+4k_{4}:k_{4}\geq 0\\}\right\\}.$

A graphical representation is provided in Figure 1.

$\mathbf{4}$$3$$2$$1$$8$$7$$6$$5$$12$$11$$10$$9$$16$$15$$14$$13$$20$$19$$18$$17$$\dots$$\dots$$\dots$$\dots$

$g^{0}_{A}$:  $g^{0}_{B}$:  $g^{0}_{C}$:

Figure 1: Initial information partitions.

Suppose the true state of the world is $4$. This means that both $A$ and $B$
know that the true state can only be either $3$ or $4$, whereas $C$ knows that
the true state can only belong to $\left\\{4,8,12,\dots\right\\}$. If everyone
publicly discloses his or her information block, each person will readily
learn that the true state is $4$. But can people find out the true state when
they disclose less than what they know? To be more concrete, suppose that each
person sends messages according to the function $f$, which is defined as
follows. For any information block $S\subseteq\mathbb{N}$,

$f(S)=\begin{cases}k&\text{if }S=\\{k\\}\text{ for some }k\in\mathbb{N}\\\
0&\text{otherwise}.\end{cases}$

In words, when someone knows that the true state is $k$, he or she sends the
message $k$, which is a shorthand for the statement “I know that the true
state is $k$”. In all other cases, the message $0$ is sent. The latter is a
shorthand for “I do not know the true state”. This formulation is a variant of
the classical puzzle of the muddy children (or unfaithful husbands or red and
white hats). See, e.g., Geanakoplos (1994).

We also assume that communication unfolds in stages. There is no limit on the
number of stages and dialogues of transfinite length are feasible. In each
stage, messages are privately exchanged. In particular, $B$ sends a message to
both $A$ and $C$, and each of $A$ and $C$ sends a message to $B$.

Let’s see what happens during the first stage of communication. Since nobody
knows the true state, everyone sends the message $0$. Clearly, nobody learns
the true state after messages are exchanged, and each person’s information
block containing $4$ stays the same. However, $B$ can refine his information
partition at the end of the first stage of communication. The reason is this.
If the true state were $5$ or $6$, $B$’s block containing the true state would
be $\\{5,6\\}$. In addition, $A$ would send the message $5$ to $B$ if the true
state were $5$, and she would send a different message, namely $0$, if the
state were $6$. In either case, $B$ would learn the true state. Formally, this
means that, at the end of the first communication stage, $B$ can split the
block $\\{5,6\\}$ in his information partition into two singletons, $\\{5\\}$
and $\\{6\\}$. Differently put, $B$ is able to discriminate between $5$ and
$6$ after one round of communication. We emphasize that this updating can be
done even when, as we are assuming, the true state is $4$. In fact, a
partition represents the information a person has not just at the true state
but at every possible state of the world. As a result, people have the
following information partitions at the end of the first stage of
communication:

$\displaystyle g^{1}_{A}$ $\displaystyle=g^{0}_{A}$ $\displaystyle g^{1}_{B}$
$\displaystyle=\left(g^{0}_{B}\backslash\\{\\{5,6\\}\\}\right)\cup\left\\{\\{5\\},\\{6\\}\right\\}$
$\displaystyle g^{1}_{C}$ $\displaystyle=g^{0}_{C}.$

In the second stage, everybody sends the message $0$. In addition, $A$ splits
her block $\\{6,9\\}$ into the two singletons $\\{6\\}$ and $\\{9\\}$. $C$
splits $\\{1+4k_{1}:k_{1}\geq 0\\}$ into $\\{5\\}$ and
$\\{1+4k_{1}:k_{1}=0\textsf{ or }k_{1}\geq 2\\}$, and splits
$\\{2+4k_{2}:k_{2}\geq 0\\}$ into $\\{6\\}$ and $\\{2+4k_{2}:k_{2}=0\textsf{
or }k_{2}\geq 2\\}$. Finally, $B$’s information partition stays the same. In
the third stage, everybody sends again the message $0$. $B$ splits
$\\{9,10\\}$ into $\\{9\\}$ and $\\{10\\}$, whereas $A$ and $C$ do not alter
their information partitions. In all subsequent stages indexed by a finite
ordinal, this process of communication and learning goes on in a similar
fashion. Everybody always sends the message $0$, and, at any stage, either $B$
alone or both $A$ and $C$ can strictly refine their information partitions.

When the stage indexed by the first limit ordinal $\omega$ is reached, each
person’s information is represented by the coarsest partition that is finer
than all the partitions he or she had at previous communication stages. More
specifically, we have:

$\displaystyle g^{\omega}_{A}$
$\displaystyle=\left\\{\\{1,2,7\\},\\{3,4\\}\right\\}\cup\left\\{\\{5+4k_{1}\\}:k_{1}\geq
0\right\\}\cup\left\\{\\{6+4k_{2}\\}:k_{2}\geq
0\right\\}\cup\left\\{\\{8+4k_{3},11+4k_{3}\\}:k_{3}\geq 0\right\\}$
$\displaystyle g^{\omega}_{B}$
$\displaystyle=\left\\{\\{1,2\\}\right\\}\cup\left\\{\\{3+4k_{1},4+4k_{1}\\}:k_{1}\geq
0\right\\}\cup\left\\{\\{5+4k_{2}\\}:k_{2}\geq
0\right\\}\cup\left\\{\\{6+4k_{3}\\}:k_{3}\geq 0\right\\}$ $\displaystyle
g^{\omega}_{C}$ $\displaystyle=\left\\{\\{1+4k_{1}\\}:k_{1}\geq
0\right\\}\cup\left\\{\\{2+4k_{2}\\}:k_{2}\geq
0\right\\}\cup\left\\{\\{3+4k_{3}:k_{3}\geq 0\\},\\{4+4k_{4}:k_{4}\geq
0\\}\right\\}.$

This means that $A$ is able to discriminate between all the states in the
first two lines of Figure 1 except for the states $1$ and $2$, which still
belong to $\\{1,2,7\\}$. Similarly, $B$ can discriminate between all the
states in the first two lines except for $1$ and $2$, which still form the
block $\\{1,2\\}$. As to $C$, she can discriminate between all the states in
the first two lines of Figure 1. Even at this stage, no one knows that the
true state of the world is $4$. Consequently, after the first limit ordinal,
everyone keeps on sending the message $0$. In the stage indexed by $\omega+1$,
$B$ splits his block $\\{1,2\\}$ into the two singletons $\\{1\\}$ and
$\\{2\\}$. In fact, if the true state were $1$ or $2$, $B$ would learn it upon
hearing from $C$ at this stage. In the next stage, $A$ splits $\\{1,2,7\\}$
into $\\{1\\}$, $\\{2\\}$ and $\\{7\\}$, and so on. At the second limit
ordinal, all three persons update their information as they did at $\omega$.
This leads to the following partitions:

$\displaystyle g^{\omega\cdot 2}_{A}$ $\displaystyle=\left(\\{\\{k\\}:k\geq
1\\}\backslash\\{\\{3\\},\\{4\\}\\}\right)\cup\left\\{\\{3,4\\}\right\\}$
$\displaystyle g^{\omega\cdot 2}_{B}$ $\displaystyle=\left(\\{\\{k\\}:k\geq
1\\}\backslash\\{\\{3\\},\\{4\\}\\}\right)\cup\left\\{\\{3,4\\}\right\\}$
$\displaystyle g^{\omega\cdot 2}_{C}$ $\displaystyle=\left\\{\\{k\\}:k\geq
1\right\\}.$

At $\omega\cdot 2$, $C$ is finally able to deduce that the true state of the
world is $4$. Both $A$ and $B$ are able to discriminate between all states
except for $3$ and $4$. Hence they do not know yet if the true state is $3$ or
$4$. In the next stage, $C$ sends the message $4$ to $B$, who then sends the
same message to $A$. Therefore, at the end of the communication stage indexed
by $\omega\cdot 2+2$, everyone has learned the true state, and the information
partitions are:

$g^{\omega\cdot 2+2}_{A}=g^{\omega\cdot 2+2}_{B}=g^{\omega\cdot
2+2}_{C}=\left\\{\\{k\\}:k\geq 1\right\\}.$ (1)

It follows that, from stage $\omega\cdot 2+2$ on, everyone will always send
the message $4$, and information partitions will clearly remain the same.

Some remarks are in order. First, we can see from (1) that everyone eventually
shares all his or her information with everyone else. This is not necessarily
true in more general cases, and the amount of information shared in a dialogue
depends on the expressive power of the function $f$.

Second, everyone agrees before the stage $\omega\cdot 2$, and everyone agrees
from $\omega\cdot 2+2$ on. That is, everyone sends the message $0$ before
$\omega\cdot 2$, and everyone sends the message $4$ from $\omega\cdot 2+2$ on.
But there is an important difference between these two cases. When everyone
agrees on $0$, there is always a state of the world at which a disagreement
would emerge if that state occurred. For example, at the first stage of the
dialogue, a disagreement would emerge if the true state were $5$. On the
contrary, when everyone agrees on $4$, a consensus is reached not just at the
true state but at every possible state of the world. As we show in Section 4,
when a consensus holds only at some states, a dialogue enables people to
refine their information partitions. On the other hand, when a consensus holds
at every possible state, a dialogue does not allow people to refine their
information, meaning that there is nothing new to be learned by exchanging
messages.

Finally, we remark that the dialogue described so far can also capture
observational learning. Specifically, suppose that, in each stage, each person
takes an action from the set $D=\mathbb{N}\cup\\{0\\}$. Stage payoffs are
determined by the following function:

$u\left(d,k\right)=\begin{cases}1&\text{if }d=k\\\ 0&\text{if }d=0\\\
-1&\text{otherwise}.\end{cases}$

In words, taking action $d\neq 0$ in state $k$ yields a reward if the action
and the state match. If they don’t, the decision maker incurs a loss. In every
state, the safe option of choosing $d=0$ is always available. There are no
payoff externalities: the payoff accruing to anyone is independent of what the
others do, and vice versa. In each stage, everyone selects an action according
to the maximin criterion. For any information block $S\subseteq\mathbb{N}$,
the chosen action $\hat{f}(S)$ is

$\hat{f}(S)=\max_{d\in D}\min_{k\in S}u(d,k).$

It is straightforward to verify that $\hat{f}(S)=f(S)$ for every $S$. Now
suppose the initial information partitions are again those represented in
Figure 1. In addition, actions are privately observed. In each stage, $B$
observes the actions chosen by $A$ and $C$, and each of $A$ and $C$ observes
$B$’s choice. Assume again that the true state is $4$. In the first stage,
everyone chooses $0$. Moreover, $B$ refines his information partition by
splitting the block $\\{5,6\\}$ into $\\{5\\}$ and $\\{6\\}$ while $A$ and $C$
do not alter their information partitions. In subsequent stages too, all three
persons update their information exactly as they did with $f$. As a
consequence, everybody learns the true state at the stage indexed by
$\omega\cdot 2+2$.

##  3 Model

###  3.1 Setup

Our object of study is a communication structure $\mathcal{C}=\langle
I,X,A,f,G\rangle$, where:

  * •

$I=\\{1,\dots,n\\}$, with $n\geq 2$, is a finite set of agents;

  * •

$X$ is a non-empty set of states of the world;

  * •

$A$ is a non-empty set of messages;

  * •

$f:\mathcal{X}\to A$ is a message function, where $\mathcal{X}$ is the set of
non-empty subsets of $X$;

  * •

$G$ is a directed graph whose set of nodes is $I$. Abusing notation, we write
$G$ to indicate both the graph and its set of edges $G\subseteq I\times I$.

Agents’ information about the true state is represented by partitions. The set
of all partitions of $X$ is $\mathcal{P}$, with typical elements $P$,
$P^{\prime}$, etc. Given a state $x\in X$ and a partition $P\in\mathcal{P}$,
the block of $P$ containing $x$ is denoted by $P(x)$. The set $\mathcal{P}$ is
partially ordered by the binary relation “is coarser than”, which is denoted
by $\leq$ and defined as follows: for any two partitions $P$ and $P^{\prime}$,
we have $P\leq P^{\prime}$ if and only if $P$ is a coarsening of $P^{\prime}$,
i.e. every block of $P$ can be written as the union of some blocks of
$P^{\prime}$. We use $P\vee P^{\prime}$ to denote the join (coarsest common
refinement) of $\\{P,P^{\prime}\\}$, and $\bigvee\left\\{P_{h}:h\in
H\right\\}$ for the join of the indexed family $\left\\{P_{h}:h\in
H\right\\}$. Similarly, we use $\bigwedge\left\\{P_{h}:h\in H\right\\}$ to
indicate the meet (finest common coarsening) of the family $\left\\{P_{h}:h\in
H\right\\}$. Recall that $\mathcal{P}$ is a complete lattice.

When agent $i$’s information is represented by $P_{i}\in\mathcal{P}$, we say
that $i$ has information $P_{i}$. Knowledge is defined in the usual way. Given
a state $x\in X$ and an event $E\subseteq X$, we say that agent $i$ knows $E$
in state $x$ if $P_{i}(x)\subseteq E$. We say that $E$ is common knowledge at
$x$ if $\bigwedge\left\\{P_{i}:i\in I\right\\}(x)\subseteq E$, where
$\bigwedge\left\\{P_{i}:i\in I\right\\}(x)$ is the unique block of the meet
containing $x$.

###  3.2 Messages, communication and learning

Agents exchange messages in a multi-stage communication process. The message
function $f$ determines how agents send messages depending on their
information. The graph $G$ determines who sends a message to whom. Both $f$
and $G$ are kept fixed as communication unfolds. We do not make any assumption
on the content of messages. In particular, we do not require messages to be
posterior beliefs about a fixed event.

#### Messages.

When $i$ has information $P_{i}$, we use the function $f_{i}:X\to A$ to
indicate what message $i$ sends at any given state $x$. We refer to $f_{i}$ as
$i$’s message function. Since no confusion should arise, we save on notation
and omit the dependence of $f_{i}$ on $P_{i}$. We follow Bacharach (1985) and
make two assumptions about message functions.

######  Assumption 1 (Like-mindedness).

For every $i\in I$, and for every partition $P_{i}\in\mathcal{P}$, if $i$ has
information $P_{i}$, then $f_{i}(x)=f(P_{i}(x))$ for every $x\in X$.

Like-mindedness captures the fact that agents share the same view of the
world. If any two agents have the same information in a given state, then they
must send the same message in that state. Consequently, if different messages
are sent, this is solely due to asymmetric information and not to, say,
different subjective states or other forms of fundamental disagreement. Notice
that, in every state $x$ and for every agent $i$, the message that $i$ sends
when $x$ occurs is a function of the smallest event that $i$ knows at $x$,
which is $P_{i}(x)$.

Another implication of Assumption 1 is that, for every $x,x^{\prime}\in X$, if
$P_{i}(x)=P_{i}(x^{\prime})$, then $f_{i}(x)=f_{i}(x^{\prime})$. This reflects
full rationality. If an agent sent different messages in different states
belonging to the same information block, then she would realize that these
states are not indistinguishable after all and so she would assign them to
different information blocks. In addition, every agent always knows the
message she is transmitting.

######  Assumption 2 (Sure thing principle (STP)).

For any $S\in\mathcal{X}$, and for any partition $\\{S_{h}:h\in H\\}$ of $S$,
if $f(S_{h})=a$ for all $h\in H$ then $f(S)=a$.

The STP is also known as union consistency222The message function
$f:\mathcal{X}\to A$ is union consistent if $S,S^{\prime}\in\mathcal{X}$,
$S\cap S^{\prime}=\emptyset$, and $f(S)=f(S^{\prime})=a$ imply $f(S\cup
S^{\prime})=a$. Strictly speaking, the union consistency of Cave (1983) and
the STP of Bacharach (1985) are equivalent only when information partitions
are finite. With possibly infinite partitions, the STP implies union
consistency, but the converse is not true. (Cave (1983)). The STP says the
following. Take any collection of mutually disjoint events $\\{S_{h}:h\in
H\\}$. If there exists a message $a\in A$ such that, for every event $S_{h}$
in the collection, an agent sends $a$ when she knows that the true state is
contained in $S_{h}$, then she must send the same message $a$ when she knows
that the true state is in $\cup_{h\in H}S_{h}$. Bacharach (1985) shows that
the STP is satisfied, among others, by expected utility maximization and by
the maximin criterion 333See Moses and Nachum (1990) for a critique of the STP
in epistemic models, and Samet (2010) and Tarbush (2016) for possible ways to
address their critique.. It is easy to verify that the message function $f$ in
the example (Section 2) satisfies the STP too. Bacharach (1985) notes that
Hurwicz’ $\alpha$-rule does not satisfy the STP when $0<\alpha<1$. We remark
that the STP might also be violated by ambiguity averse agents. Specifically,
suppose an agent has full information and, at every block of her information
partition, she announces that her optimal choice is $a$. Under ambiguity
aversion, the agent could choose a different action $b$ if she only knew that
the state is contained in the union of those blocks444I thank an anonymous
reviewer for suggesting that the STP does not hold under ambiguity aversion..

We can now introduce working partitions (Weyers (1992)). Given a message
function $f_{i}$, we use $W_{i}$ to denote the corresponding working
partition. For every $x\in X$, the block of $W_{i}$ containing $x$ is
$W_{i}(x):=\\{x^{\prime}\in X:f_{i}(x^{\prime})=f_{i}(x)\\}$. In words,
$W_{i}(x)$ is the event “$i$ sends message $a$” for some $a\in A$. Therefore,
we can interpret $W_{i}(x)$ as the information that $i$ conveys to any
receiver $j\neq i$ by sending message $f_{i}(x)$ in state $x$. The fact that
$W_{i}$ is a partition reflects the lack of ambiguity about the interpretation
of messages. Since no confusion should arise, we save on notation and omit the
dependence of $W_{i}$ on the underlying information partition $P_{i}$.
Finally, notice that $W_{i}$ is necessarily a coarsening of $P_{i}$.

#### Communication and learning.

Communication unfolds in stages. We do not put any limit on the length of the
communication process. In particular, agents can engage in dialogues of
transfinite length. A transfinite protocol is a sequence
$\left(G^{\alpha}:\alpha\in\mathsf{Ord}\right)$, where $G^{\alpha}\subseteq
I\times I$ for every ordinal $\alpha$. If $(i,j)\in G^{\alpha}$ for some
ordinal $\alpha$, there is a directed edge from $i$ to $j$ and this means that
$i$ sends a message to $j$ in stage $\alpha$. Throughout the paper, we confine
ourselves to protocols such that $G^{\alpha}=G$ for every ordinal $\alpha$. In
other words, we consider only stationary protocols induced by the fixed graph
$G$.

We are going to use a function $g$ to describe how receivers update their
information upon receiving messages. Before formally defining such a function,
we introduce a couple of auxiliary objects. Let $\mathcal{P}^{n}$ be the
$n$-fold Cartesian product of $\mathcal{P}$. An element of $\mathcal{P}^{n}$
is an indexed collection $\boldsymbol{P}=\left(P_{1},\dots,P_{n}\right)$ of
partitions of $X$. The set $\mathcal{P}^{n}$ is endowed with the following
product order:

$(P_{1},\dots,P_{n})\leq(P^{\prime}_{1},\dots,P^{\prime}_{n})\iff P_{i}\leq
P^{\prime}_{i}\;\text{ for all }i\in I.$

Notice that $\mathcal{P}^{n}$ is a complete lattice too. For each $i\in I$,
the (possibly empty) set $S(i):=\left\\{j\in I:(j,i)\in G\right\\}$ is the
subset of agents who send a message to $i$. We are now ready to define the
function $g:\mathcal{P}^{n}\to\mathcal{P}^{n}$ as follows:

$g_{i}\left((P_{1},\dots,P_{n})\right)=\begin{cases}\bigvee\left\\{P_{i}\vee
W_{j}\right\\}_{j\in S(i)}&\text{if }S(i)\neq\emptyset\\\
P_{i}&\text{otherwise},\end{cases}$ (2)

where we write $g_{i}\left((P_{1},\dots,P_{n})\right)$ to denote the $i$th
component of $g\left((P_{1},\dots,P_{n})\right)$.

In words, $g$ maps profiles of information partitions at the beginning of a
given stage of communication to profiles of updated partitions. The updating
is done in light of the messages exchanged during that communication stage. If
$i$ does not receive any message, her information stays the same. But if she
receives a message from $j$, she updates her information by taking the join
between her own information partition $P_{i}$ and $j$’s working partition
$W_{j}$. If $i$ receives messages from more than one sender, she takes the
join between her own information partition and the working partitions of all
$j\in S(i)$. The learning process just described is introduced by Parikh and
Krasucki (1990) and then amended by Weyers (1992). In particular, Weyers
(1992) argues that fully rational agents update not only the partition block
containing the true state of the world but their whole information partition.

Given a graph $G$ and a profile $\boldsymbol{P^{0}}$ of initial information
partitions, we define the dialogue starting from $\boldsymbol{P^{0}}$ as the
sequence $\left(g^{\alpha}:\alpha\in\mathsf{Ord}\right)$ constructed
recursively as follows:

$\displaystyle g^{0}$ $\displaystyle:=\boldsymbol{P^{0}},$ $\displaystyle
g^{\alpha+1}$ $\displaystyle:=g\left(g^{\alpha}\right)\text{ for every ordinal
}\alpha,$ $\displaystyle g^{\lambda}$
$\displaystyle:=\bigvee\left\\{g^{\alpha}:\alpha<\lambda\right\\}\text{ for
every limit ordinal }\lambda.$

In words, a dialogue is a transfinite sequence in $\mathcal{P}^{n}$ starting
from an initial profile $\boldsymbol{P^{0}}$ and constructed by iterating the
function $g$ induced by $G$. For every ordinal $\alpha$, the element
$g^{\alpha}$ of the dialogue tells us what information agents have at the end
of the $\alpha$th stage of communication. At every successor ordinal, agents
update their information as per (2). At every limit ordinal $\lambda$, each
agent takes the join of the information partitions she had at all the previous
stages $\alpha<\lambda$ of the dialogue.

The initial profile $\boldsymbol{P^{0}}$ can be thought of as exogenous
information, whereas partitions $g^{\alpha}$, with $\alpha>0$, can be
interpreted as endogenous information. Differently put, $\boldsymbol{P^{0}}$
captures the information conveyed by a privately observed signal about the
state. We do not explicitly model this signal and only assume that it induces
partitional information. Nature acts only once and chooses the true state of
the world $x$, which in turn determines the signal realization $P_{i}^{0}(x)$
observed by each $i$. Subsequent information partitions are formed as the
above process of communication and learning unfolds.

We define a dialogue as a sequence of partitions and not as a sequence of
messages. This allows us to exploit the lattice structure of $\mathcal{P}^{n}$
when we study dialogues leading to consensus in Section 4. Notice that, for a
given message function $f$, a profile $\boldsymbol{P}$ of information
partitions fully determines the unique profile of messages transmitted at
every state. We conclude this section by emphasizing that the whole structure
of the model is common knowledge. In particular, it is commonly known who
talks with whom and when, how partition blocks are mapped to messages, and how
information is updated.

##  4 Results

In this section we study conditions under which a rational dialogue eventually
leads to consensus. We first give a full characterization of consensus for the
static case, i.e. for a fixed profile of information partitions; we then move
on to the dynamic case, in which communication is allowed.

###  4.1 Consensus

Let $\boldsymbol{P}\in\mathcal{P}^{n}$ be a profile of information partitions
at a given stage. We say that a consensus holds at that stage if, for all
$i,j\in I$, we have that $f_{i}=f_{j}$. That is, for all $i,j\in I$,
$f_{i}(x)=f_{j}(x)$ for every state $x\in X$. If agents agree at some state
$x$ but not necessarily at every state, then we say that a partial consensus
holds at $x$. In the example in Section 2, the consensus that holds at the
true state $4$ is only partial before the stage $\omega\cdot 2$, but then it
becomes a full consensus from $\omega\cdot 2+2$ on.

Our first result is a characterization of consensus for a given profile of
information partitions.

######  Proposition 1.

Suppose agents have information
$\boldsymbol{P}=\left(P_{1},\dots,P_{n}\right)$. Then the following are
equivalent:

  1. a)

For all $x\in X$, the profile of messages that is sent at $x$, i.e. the event

$E(x)=\left\\{x^{\prime}\in
X:f_{1}(x^{\prime})=f_{1}(x),\dots,f_{n}(x^{\prime})=f_{n}(x)\right\\},$

is common knowledge at $x$;

  2. b)

For all $i,j\in I$, $f_{i}=f_{j}$;

  3. c)

For all $i,j\in I$, $W_{i}=W_{j}$.

###### Proof.

Firstly, $a)\implies b)$ follows from Theorem 3 in Bacharach (1985). Secondly,
$b)\implies c)$ follows immediately from the definition of working partitions.
Finally, in order to show $c)\implies a)$, fix a state $x\in X$. By the
definition of working partitions, for every $i\in I$, the event
$\left\\{x^{\prime}\in X:f_{i}(x^{\prime})=f_{i}(x)\right\\}$ is the same as
$W_{i}(x)$. Let $W(x):=\cap_{i\in I}W_{i}(x)$. Since every $W_{i}$ is a
coarsening of $P_{i}$, and since $W_{i}(x)=W_{j}(x)$ for all $i,j\in I$ by
assumption, we have that, for all $i\in I$,

$P_{i}(x)\subseteq W_{i}(x)=W(x).$

Therefore, for all $i\in I$,

$P_{i}(x)\subseteq\bigwedge\left\\{P_{i}:i\in I\right\\}(x)\subseteq W(x).$

∎

Proposition 1 says that consensus is equivalent to an epistemic configuration
where, at every state, the profile of messages that are being sent at that
state is common knowledge. Furthermore, when messages are commonly known, they
must be the same. The latter statement is nothing other than the generalized
version of Aumann’s agreement theorem established in Bacharach (1985). We can
also say that a consensus cannot hold without it being common knowledge that
it holds. This is not necessarily true for partial consensus. In the example
(Section 2), the profile of messages sent at the true state is not common
knowledge until a full consensus is reached at the stage $\omega\cdot 2+2$.

We remark that consensus does not imply that all agents have the same
information partitions. Furthermore, the equivalence in Proposition 1
crucially relies on the STP. As we show in Section 5, if the STP does not
hold, one can only conclude that $b)\implies c)\implies a)$. Consequently, it
would no longer be impossible to agree to disagree.

The following corollary will prove useful in establishing subsequent results.

######  Corollary 1.

If $f_{i}\neq f_{j}$, then $P_{i}<P_{i}\vee W_{j}$ or $P_{j}<P_{j}\vee W_{i}$.

###### Proof.

By contraposition, suppose that neither $P_{i}<P_{i}\vee W_{j}$ nor
$P_{j}<P_{j}\vee W_{i}$ hold. Since it is always the case that $P_{i}\leq
P_{i}\vee W_{j}$ and $P_{j}\leq P_{j}\vee W_{i}$, we must have both
$P_{i}=P_{i}\vee W_{j}$ and $P_{j}=P_{j}\vee W_{i}$. This implies that $W_{i}$
is a coarsening of $P_{j}$ and $W_{j}$ is a coarsening of $P_{i}$. Combining
this with the fact that each working partition is a coarsening of the
underlying information partition, we have that both $W_{i}$ and $W_{j}$ are
common coarsenings of $P_{i}$ and $P_{j}$. Therefore, for every $x\in X$,
$W_{i}(x)\cap W_{j}(x)$ is common knowledge at $x$ between $i$ and $j$. Thus
it follows from Proposition 1 that $f_{i}=f_{j}$. ∎

The interpretation is that, if $i$ and $j$ disagree at some state, then it
must be the case that either $i$ can strictly refine her information by
receiving a message from $j$, or $j$ can strictly refine her information by
receiving a message from $i$, or both. In other words, when two agents
disagree, at least one of them can learn some new information from the other.

###  4.2 Dialogues leading to consensus

We now examine conditions under which dialogues lead to consensus. Given a
communication graph $G$ and a profile $\boldsymbol{P^{0}}$ of initial
information partitions, we say that the dialogue
$\left(g^{\alpha}:\alpha\in\mathsf{Ord}\right)$ induced by $G$ and starting
from $\boldsymbol{P^{0}}$ leads to a consensus if, for every $i,j\in I$, we
have that $f_{i}^{\alpha}=f_{j}^{\alpha}$ for some ordinal $\alpha$, where
$f_{i}^{\alpha}$ is $i$’s message function associated with $g_{i}^{\alpha}$.
Differently put, a dialogue leads to consensus if the sequence
$\left(g^{\alpha}:\alpha\in\mathsf{Ord}\right)$ contains a profile
$g^{\alpha}\in\mathcal{P}^{n}$ at which everybody agrees.

It is clear that the properties of $G$ affect the function $g$ and,
consequently, the sequence $\left(g^{\alpha}:\alpha\in\mathsf{Ord}\right)$. We
make the following preliminary observation.

######  Remark.

For any $G$, the function $g$ is inflationary but need not be monotone555Let
$P$ be a poset. A function $h:P\to P$ is monotone (or order-preserving) if
$x\leq y\implies h(x)\leq h(y)$. We say that $h$ is inflationary (or
increasing) if, for all $x\in P$, $x\leq h(x)$..

###### Proof.

It follows immediately from (2) that $g$ is inflationary. The following
example shows that $g$ need not be monotone. Let $X=\left\\{x,y,w,z\right\\}$,
$I=\\{1,2\\}$, and let the message function $f$ be such that
$f(\\{x\\})=f(\\{x,y\\})=a$, and $f(S)=b$ for any other non-empty subset $S$
of $X$. In addition, suppose the communication graph is $G=\\{(1,2),(2,1)\\}$.
Now take the following elements of $\mathcal{P}^{2}$:

$\displaystyle\boldsymbol{P}$
$\displaystyle=\left(P_{1},P_{2}\right)=\left(\\{X\\},\\{\\{x,y\\},\\{w,z\\}\\}\right)$
$\displaystyle\boldsymbol{P^{\prime}}$
$\displaystyle=\left(P^{\prime}_{1},P^{\prime}_{2}\right)=\left(\\{X\\},\\{\\{x\\},\\{y\\},\\{w\\},\\{z\\}\\}\right).$

Thus we have

$\displaystyle g\left(\boldsymbol{P}\right)$
$\displaystyle=\left(\\{\\{x,y\\},\\{w,z\\}\\},\\{\\{x,y\\},\\{w,z\\}\\}\right)$
$\displaystyle g\left(\boldsymbol{P^{\prime}}\right)$
$\displaystyle=\left(\\{\\{x\\},\\{y,w,z\\}\\},\\{\\{x\\},\\{y\\},\\{w\\},\\{z\\}\\}\right).$

Therefore, $\boldsymbol{P}\leq\boldsymbol{P^{\prime}}$ but
$g\left(\boldsymbol{P}\right)\not\leq g\left(\boldsymbol{P^{\prime}}\right)$.
∎

Since $g$ is inflationary, the sequence $(g^{\alpha}:\alpha\in\mathsf{Ord})$
is increasing. That is, for every $\alpha,\beta\in\mathsf{Ord}$, we have that
$\beta<\alpha$ implies $g^{\beta}\leq g^{\alpha}$. This can easily be proved
by induction on $\alpha$.

We now introduce the following two subsets of $\mathcal{P}^{n}$:

$\mathsf{Cons}(f):=\left\\{\boldsymbol{P}\in\mathcal{P}^{n}:f_{i}=f_{j}\text{
for all }i,j\in I\right\\}$

and

$\mathsf{Fix}(g):=\left\\{\boldsymbol{P}\in\mathcal{P}^{n}:g(\boldsymbol{P})=\boldsymbol{P}\right\\}.$

In words, $\mathsf{Cons}(f)$ is the set of partition profiles at which a
consensus holds, whereas $\mathsf{Fix}(g)$ is the set of fixed points of $g$.

######  Proposition 2.

For any $G$, we have $\emptyset\neq\mathsf{Cons}(f)\subseteq\mathsf{Fix}(g)$.

###### Proof.

To show that $\mathsf{Cons}(f)$ is non-empty, take any profile
$\left(P_{1},\dots,P_{n}\right)\in\mathcal{P}^{n}$ such that $P_{i}=P_{j}$ for
every $i,j\in I$. By like-mindedness, all such profiles are contained in
$\mathsf{Cons}(f)$.

To show the inclusion $\mathsf{Cons}(f)\subseteq\mathsf{Fix}(g)$, take
$\left(P_{1},\dots,P_{n}\right)\in\mathsf{Cons}(f)$. By Proposition 1,
$\left(P_{1},\dots,P_{n}\right)$ is such that $W_{i}=W_{j}$ for any $i,j\in
I$. Therefore, since $W_{i}$ is a coarsening of $P_{i}$, we have that
$P_{i}\vee W_{j}=P_{i}$ for any $i,j\in I$. Thus
$\left(P_{1},\dots,P_{n}\right)\in\mathsf{Fix}(g)$. ∎

Since $g$ is inflationary, the non-emptiness of $\mathsf{Fix}(g)$ can also be
proved by invoking the Bourbaki-Witt fixed point theorem.

Now, it is clear that if a dialogue
$\left(g^{\alpha}:\alpha\in\mathsf{Ord}\right)$ contains a fixed point at
$g^{\alpha}$, then it stays constant at any $\beta>\alpha$. However, it is not
necessarily the case that such a dialogue leads to consensus. In order for
this to be the case, we need to make sure that the communication process
induced by $G$ is sufficiently rich. We thus make the following assumption.

######  Assumption 3.

The communication graph $G$ contains a spanning subgraph666Recall that a
spanning subgraph of $G$ is a subgraph $G^{\prime}\subseteq G$ with the same
set of vertexes as $G$. $G^{\prime}$ such that:

  1. a)

$G^{\prime}$ is strongly connected: for every distinct $i,j\in I$, there
exists a directed path in $G^{\prime}$ from $i$ to $j$ and a directed path
from $j$ to $i$;

  2. b)

$G^{\prime}$ is symmetric: for every $i,j\in I$, if $(i,j)\in G^{\prime}$,
then $(j,i)\in G^{\prime}$.

Strong connectedness says that no one is excluded from communication, i.e.
everyone communicates with everybody else, either directly or indirectly.
Symmetry means that communication is reciprocal. When these two conditions are
met, the following equivalence holds.

######  Proposition 3.

If $G$ satisfies Assumption 3, then $\mathsf{Cons}(f)=\mathsf{Fix}(g)$.

###### Proof.

By Proposition 2, it is enough to show that
$\mathsf{Fix}(g)\subseteq\mathsf{Cons}(f)$. Let
$\boldsymbol{P}=\left(P_{1},\dots,P_{n}\right)$ be a fixed point of $g$.
Suppose by way of contradiction that $\boldsymbol{P}$ does not induce a
consensus. Hence there are distinct $i$ and $j$ in $I$ such that $f_{i}\neq
f_{j}$. By strong connectedness, there exists a directed path in
$G^{\prime}\subseteq G$ from $i$ to $j$: that is, for some integer $K\geq 1$,
there is a path $i_{0},i_{1},\dots,i_{K}$ in $G^{\prime}$ such that $i_{0}=i$
and $i_{K}=j$. Since $i$ and $j$ disagree, this path must contain an edge
$(i_{k},i_{k+1})$ such that $i_{k}$ and $i_{k+1}$ disagree, for some
$k\in\\{0,\dots,K-1\\}$. By symmetry, $(i_{k+1},i_{k})\in G^{\prime}$. By
Corollary 1, we have

$P_{k}<P_{k}\vee W_{k+1}\;\text{ or }\;P_{k+1}<P_{k+1}\vee W_{k},$

and using this in (2) we obtain

$P_{k}<g_{k}(\boldsymbol{P})\;\text{ or }\;P_{k+1}<g_{k+1}(\boldsymbol{P}),$

so contradicting the hypothesis that $\boldsymbol{P}$ is a fixed point of $g$.
∎

In light of Proposition 3, looking for dialogues leading to consensus is the
same as looking for fixed points of $g$. Intuitively, we know from Corollary 1
that, in case of disagreement between $i$ and $j$, learning can take place in
either direction. Assumption 3 makes sure that communication between $i$ and
$j$ is reciprocal, so that it can never be the case that $i$ and $j$ disagree
without having the possibility of exchanging messages between each other. The
importance of reciprocity in communication is pointed out by Krasucki (1996),
and Example 2 in Parikh and Krasucki (1990) shows how a consensus may never
emerge if one dispenses with it.

We are now ready to state our main result. In what follows, we write
$\alpha^{*}$ to denote the least ordinal $\alpha$ such that
$g^{\alpha+1}=g^{\alpha}$.

######  Theorem.

Let $\mathcal{C}=\langle I,X,A,f,G\rangle$ be a communication structure
satisfying Assumptions 1-3. For any profile $\boldsymbol{P^{0}}$ of initial
information partitions, the dialogue
$\left(g^{\alpha}:\alpha\in\mathsf{Ord}\right)$ induced by $G$ and starting
from $\boldsymbol{P^{0}}$ leads to a consensus. Furthermore, $|\alpha^{*}|\leq
n|X|$.

###### Proof.

Since $g$ is inflationary, and since $\mathcal{P}^{n}$ is a complete lattice,
by following the same argument as in the proof of Theorem 12.9 in Roman (2008)
we have that the sequence $\left(g^{\alpha}:\alpha\in\mathsf{Ord}\right)$
starting from $\boldsymbol{P^{0}}$ is always well-defined, increasing, and
contains one, and only one, fixed point of $g$. By Proposition 3, the dialogue
$\left(g^{\alpha}:\alpha\in\mathsf{Ord}\right)$ induces a consensus.

In order to show that $|\alpha^{*}|\leq n|X|$, take the subsequence
$\left(g^{\alpha}:\alpha\leq\alpha^{*}\right)$. Since
$\left(g^{\alpha}:\alpha\in\mathsf{Ord}\right)$ is increasing and
$g^{\alpha^{*}}$ is a fixed point of $g$, the subsequence
$\left(g^{\alpha}:\alpha\leq\alpha^{*}\right)$ is strictly increasing, i.e.
for all $\alpha,\beta\leq\alpha^{*}$,

$\beta<\alpha\implies g^{\beta}\leq g^{\alpha}\;\text{ and }\;g^{\beta}\neq
g^{\alpha}.$ (3)

Now define the image of $\left(g^{\alpha}:\alpha\leq\alpha^{*}\right)$ as
$\mathsf{Im}:=\left\\{g^{\alpha}:\alpha\leq\alpha^{*}\right\\}$. Since
$\left(g^{\alpha}:\alpha\leq\alpha^{*}\right)$ is strictly increasing,
$\mathsf{Im}$ is a well-ordered subset of $\mathcal{P}^{n}$ having order type
$\alpha^{*}+1$. Furthermore, for every $i\in I$, let
$\mathsf{Im}_{i}:=\left\\{g_{i}^{\alpha}:\alpha\leq\alpha^{*}\right\\}$ be the
$i$th projection of $\mathsf{Im}$. Notice that $\mathsf{Im}_{i}$ is a well-
ordered chain in $\mathcal{P}$ having order type at most $\alpha^{*}+1$. By
Lemma 3.1 in Avery et al. (2018), we have that $|\mathsf{Im}_{i}|\leq|X|$. Now
construct a map $\phi:\alpha^{*}+1\to\bigsqcup_{i=1}^{n}\mathsf{Im}_{i}$,
where $\bigsqcup$ denotes disjoint union, as follows. Let
$\phi(0):=P^{0}_{1}$, and for every $0<\alpha\leq\alpha^{*}$,

$\phi(\alpha):=\min_{i}\left\\{g_{i}^{\alpha}:g_{i}^{\beta}\leq
g_{i}^{\alpha}\;\text{ and }\;g_{i}^{\beta}\neq g_{i}^{\alpha}\;\text{ for all
}\;\beta<\alpha\right\\}.$ (4)

In words, $\phi$ maps each ordinal $\alpha$ less than or equal to $\alpha^{*}$
to an individual partition $g_{i}^{\alpha}$ that is a strict refinement of all
partitions $g_{i}^{\beta}$ having index less than $\alpha$. Without loss of
generality, in case of multiple individual partitions satisfying (4), we take
the one with the lowest (agent) index. It follows from (3) that $\phi$ is
well-defined and injective. Therefore we have

$|\alpha^{*}|\leq|\alpha^{*}+1|\leq\Big{|}\bigsqcup_{i=1}^{n}\mathsf{Im}_{i}\Big{|}\leq
n|X|.$

∎

Clearly, when $X$ is an infinite set, we have $|\alpha^{*}|\leq n|X|=|X|$.

##  5 Discussion

###  5.1 Related literature

The paper contributes to the vast literature on common knowledge and agreement
initiated by Aumann (1976), surveys of which can be found in Bonanno and
Nehring (1997) and (Ménager, 2006b, Chapter 2). In particular, we contribute
to the strand that investigates rational dialogues and convergence to
consensus. Krasucki (1996) and Parikh (1992) are the closest papers, and below
we briefly discuss the connection with them.

Geanakoplos and Polemarchakis (1982) introduce dialogues in a two-agent model
with a finite state space where the messages exchanged are posterior beliefs
about a fixed event. Bacharach (1985) and Cave (1983) show that a consensus
can be reached not only when people communicate posterior beliefs but also
when they communicate the values of any function satisfying a condition akin
to the sure thing principle from decision theory. Bacharach (1985) considers a
two-agent model in which initial information partitions are finite, whereas
Cave (1983) assumes that the state space is a measure space and that
communication is public. Nielsen (1984), Washburn and Teneketzis (1984) and
Bergin (1989) study convergence to consensus but they all confine their
attention to the case where the state space is a probability space.

Dialogues with private communication between more than two agents are
introduced by Parikh and Krasucki (1990) and further examined in Krasucki
(1996), Heifetz (1996) and Ménager (2006a). They all consider models with
finite information partitions. The model in our paper is the same as that in
Krasucki (1996) except for two crucial aspects. While he considers finite
partitions and dialogues of finite length, we do not make any of these
finiteness assumptions.

Dialogues of transfinite length are studied in Aumann and Hart (2003) and
Parikh (1992). Contrary to this paper, Aumann and Hart (2003) consider
problems in which a simultaneous-move game is played after countably many
cheap talk messages have been exchanged. Like this paper, Parikh (1992)
studies common knowledge acquisition and consensus. However, there are
substantial differences with our paper. Most importantly, his model uses
Kripke structures and has two agents and a state space that is at most
countably infinite. In our paper, we use Aumann (or partition) structures, we
have many agents, and we allow the state space to have any cardinality. We
remark that, while there is scant literature on transfinite dialogues, there
are many papers in which the analysis of interactive knowledge is done on the
transfinite domain but without any form of communication between agents. For
example, see Lipman (1994) and Heifetz and Samet (1998).

Finally, this paper is also related to Mueller-Frank (2013), in which a
framework for learning in social networks is provided. The main difference
with our paper is that only dialogues having order type $\omega$ are
considered by Mueller-Frank (2013). As a consequence, his notion of
convergence when information partitions are infinite is different from ours.

###  5.2 The Sure Thing Principle

Proposition 1 hinges upon the STP. If one dispenses with it, the equivalence
$a)\iff b)\iff c)$ breaks down and one can only conclude that $b)\implies
c)\implies a)$. Notice that we do not use the STP when we show that
$b)\implies c)$ and $c)\implies a)$ in the proof of Proposition 1. In the
following two examples, we show that the converse implications do not
necessarily hold. First, let us show that $b)$ does not follow from $a)$ or
from $c)$. Set $X=\left\\{x,y\right\\}$ and let the message function be such
that $f(\\{x\\})=f(\\{y\\})=a$ and $f(\\{x,y\\})=b$. Suppose that there are
two agents whose information partitions are
$P_{1}=\left\\{\\{x\\},\\{y\\}\right\\}$ and
$P_{2}=\left\\{\\{x,y\\}\right\\}$. At every state, it is common knowledge
what messages $1$ and $2$ are sending, but clearly $f_{1}\neq f_{2}$. Notice
that we also have $W_{1}=W_{2}=\\{X\\}$.

Now we show that $c)$ does not follow from $a)$. Set $X=\\{x,y,z\\}$ and let
the message function be such that

$\displaystyle f(\\{x\\})=f(\\{y\\})$ $\displaystyle=a$ $\displaystyle
f(\\{z\\})=f(\\{x,y\\})$ $\displaystyle=b.$

Suppose there are two agents with the following information partitions:

$\displaystyle P_{1}$ $\displaystyle=\left\\{\\{x\\},\\{y\\},\\{z\\}\right\\}$
$\displaystyle P_{2}$ $\displaystyle=\left\\{\\{x,y\\},\\{z\\}\right\\}.$

At every state, the profile of messages is common knowledge but $W_{1}\neq
W_{2}$.

###  5.3 Common knowledge of the communication protocol

Our analysis assumes that the way in which a dialogue unfolds is common
knowledge. In particular, the communication protocol induced by the graph $G$
is commonly known, and this is crucial in order to have a well-defined
learning process. When $i$ receives a message from $j$, she knows exactly whom
$j$ talked with in the past and, consequently, she can infer what information
$j$ learned from that history even if she does not necessarily know the actual
message that $j$ sent to others or received in some state. Differently put, a
commonly known communication protocol implies that the informational content
of any given message is not ambiguous, so making it possible for people to
learn. In the case in which such a common knowledge assumption is relaxed,
learning is not well-defined in our framework and convergence to consensus is
not guaranteed. In these cases, Koessler (2001) and Tsakas and Voorneveld
(2011) show that one needs to enlarge the state space so as to include any
possible history of communication. In so doing, uncertainty about the
communication structure can be dealt with in the enlarged state space. The
fact that we keep the state space fixed throughout a dialogue is a direct
consequence of having a commonly known communication protocol.

We also emphasize that communication takes place through a faultless and fully
reliable channel. To wit, when $i$ sends a message to $j$, the message is
delivered to $j$ with absolute certainty, and it is common knowledge that it
is so. We rule out the possibility that a message never reaches the intended
recipient and also the possibility that a recipient gets a different message
than what was sent by the sender. Should the communication channel be
unreliable, we would be in a situation akin to the email game of Rubinstein
(1989), where convergence to consensus is not guaranteed to hold.

###  5.4 Non-strategic communication

We assume that communication is not strategic. When strategic motives are
introduced, a consensus is not always reached. In a different yet related
setting, Anderlini et al. (2011) show that, while agents with common interests
are able to aggregate their information in a full learning equilibrium, no
such an equilibrium can be sustained when interests diverge. However,
Ostrovsky (2012)777I thank an anonymous reviewer for bringing my attention to
this paper. shows that, for a specific class of securities, strategic
incentives do not preclude traders in dynamic financial markets from
aggregating their private information and reaching a consensus.

###  5.5 Necessity of Assumption 3

Strong connectedness and symmetry in Assumption 3 are sufficient for the
emergence of consensus in a dialogue but they are not necessary. For example,
consider an initial profile $\boldsymbol{P^{0}}$ such that the individual
partitions are ordered as follows: $P^{0}_{1}\leq P^{0}_{2}\leq\cdots\leq
P^{0}_{n}$. A consensus can be obtained after only one round of communication
by letting $n$, who is the most informed agent, send a message to everybody
else.

###  5.6 Information sharing

When a consensus is reached, agents do not always have the same information.
But if the message function has enough expressive power, then everyone ends up
having the same information when a consensus holds. More specifically, suppose
the message function $f$ is injective888I thank an anonymous reviewer for
suggesting the connection between injective message functions and information
sharing.. Clearly, the STP is vacuously satisfied. Moreover, each agent’s
information partition $P_{i}$ is the same as her working partition $W_{i}$.
And since everyone has the same working partition when a consensus is reached,
everyone must have the same information partition too.

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