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  <front>
    <journal-meta />
    <article-meta>
      <title-group>
        <article-title>Value-driven Collaborative Privacy Decision Making</article-title>
      </title-group>
      <contrib-group>
        <contrib contrib-type="author">
          <string-name>Francesca Mosca</string-name>
          <xref ref-type="aff" rid="aff0">0</xref>
        </contrib>
        <contrib contrib-type="author">
          <string-name>Jose M. Such</string-name>
          <xref ref-type="aff" rid="aff0">0</xref>
        </contrib>
        <contrib contrib-type="author">
          <string-name>Peter McBurney</string-name>
          <email>peter.mcburneyg@kcl.ac.uk</email>
          <xref ref-type="aff" rid="aff0">0</xref>
        </contrib>
        <aff id="aff0">
          <label>0</label>
          <institution>Department of Informatics King's College London</institution>
        </aff>
      </contrib-group>
      <abstract>
        <p>Multiparty privacy conflicts (MPCs) occur when the privacy of a group of individuals is affected by the same piece of information, yet they have different (possibly conflicting) individual privacy preferences. One of the domains in which MPCs manifest strongly is online social networks, where the majority of users reported having suffered MPCs when sharing photos in which multiple users were depicted. Previous work on supporting users to make collaborative decisions to decide on the optimal sharing policy to prevent MPCs share one critical limitation: they lack transparency in terms of how the optimal sharing policy recommended was arrived at, which has the problem that users may not be able to comprehend why a particular sharing policy might be the best to prevent a MPC, potentially hindering adoption and decreasing the chance for users to accept or influence the recommendations. In this paper, we report our work in progress towards an AI-based model for collaborative privacy decision making that can justify its choices and allows users to influence them based on human values. In particular, the model considers both the individual privacy preferences of the users involved as well as their values to drive the negotiation process to arrive at an agreed sharing policy. We formally prove that the model we propose is correct, complete and that it terminates in finite time. We also provide an overview of the future directions in this line of research.</p>
      </abstract>
    </article-meta>
  </front>
  <body>
    <sec id="sec-1">
      <title>Introduction</title>
      <p>
        Collaborative platforms like online social networks (OSNs),
cloud-based file storage and sharing, cloud-based
collaborative documents, and so on, proved particularly challenging
for users to manage privacy and access to their data,
especially when that data affects multiple users at the same
time
        <xref ref-type="bibr" rid="ref19 ref28 ref28">(Such and Criado 2018; Paci, Squicciarini, and
Zannone 2018)</xref>
        . Whenever documents regard multiple users, e.g.
internal files of a company, pictures on social networks, or
working sheets on online sharing platforms, the privacy
settings and the sharing rights should be understood and agreed
by all the users involved. If this does not happen, issues
deriving from access control are likely to arise and no system
has yet been implemented in order to detect them in advance
or solve them efficiently once they occur
        <xref ref-type="bibr" rid="ref19 ref28 ref28">(Such and Criado
2018; Paci, Squicciarini, and Zannone 2018)</xref>
        .
      </p>
      <p>
        A multi-party privacy conflict (MPC) is defined as a
situation where more than one person is involved in some
content (people appearing in a picture together, co-authors of
a document, etc.) and they disagree on the level of
sharing/privacy to be assigned to that content before sharing
it online. In particular, MPCs on online social networks
(OSNs) have received some attention in recent literature
        <xref ref-type="bibr" rid="ref1 ref14 ref15 ref17 ref28 ref3 ref30 ref32 ref38 ref5 ref7">(Besmer and Richter Lipford 2010; Lampinen et al. 2011;
Wisniewski, Lipford, and Wilson 2012; Acquisti,
Brandimarte, and Loewenstein 2015; Liang et al. 2015; Acquisti
et al. 2017; Such et al. 2017; Such and Criado 2018)</xref>
        .
Nowadays on most platforms it is possible only for the uploaders
to define the privacy settings of photographic contents; if
another person who is depicted in the picture (a co-owner of
the picture) disagrees with such setting, there is no efficient
support from the platform side
        <xref ref-type="bibr" rid="ref38">(Wisniewski, Lipford, and
Wilson 2012)</xref>
        : the co-owners can flag the picture or contact
the uploader to ask for its removal, but the damage may have
already been caused, as the item is potentially visualised by
many before any counteraction is made. So, reparative
solutions are in general not enough
        <xref ref-type="bibr" rid="ref17 ref30 ref32">(Such et al. 2017)</xref>
        .
      </p>
      <p>
        Pushed by the need of improving the support to the user
provided by the collaborative platforms and taking
advantage of the knowledge gathered by previous studies on
usable privacy towards helping users making privacy choices
online
        <xref ref-type="bibr" rid="ref1">(Acquisti et al. 2017)</xref>
        and on empirical evidence about
MPCs in practice
        <xref ref-type="bibr" rid="ref17 ref30 ref32 ref38">(Wisniewski, Lipford, and Wilson 2012;
Such et al. 2017)</xref>
        , our main contribution is the definition of
a model that (i) invites the users to interact between
themselves in accordance with their values, in order to select
the best collective sharing policy to prevent and/or resolve
a MPC, and (ii) is able to justify its suggestion based on the
values of those users involved, helping users understand the
properties of the suggested action and the limitations of the
discarded ones. In particular, in this paper we propose our
theoretical model, which follows knowledge-based AI
techniques and the theory of basic values
        <xref ref-type="bibr" rid="ref22">(Schwartz 2010)</xref>
        to
support users while collaboratively deciding on a common
sharing policy to resolve an MPC. According to the
typology of dialogues of
        <xref ref-type="bibr" rid="ref36">(Walton and Krabbe 1995)</xref>
        , our model
is a negotiation dialogue, because the participants have
potentially conflicting goals, which may not be satisfied
simultaneously
        <xref ref-type="bibr" rid="ref16">(McBurney and Parsons 2009)</xref>
        . Therefore, we
particularly based on AI-techniques like negotiation
frameworks
        <xref ref-type="bibr" rid="ref8">(Fatima, Kraus, and Wooldridge 2014)</xref>
        , which have
already been applied to other privacy domains
        <xref ref-type="bibr" rid="ref17 ref30 ref32">(Such 2017)</xref>
        .
      </p>
    </sec>
    <sec id="sec-2">
      <title>Related Work</title>
      <p>
        We refer the reader to
        <xref ref-type="bibr" rid="ref28">(Such and Criado 2018)</xref>
        for a
comprehensive review of literature on MPCs in social media and to
        <xref ref-type="bibr" rid="ref19 ref28">(Paci, Squicciarini, and Zannone 2018)</xref>
        for a more general
survey on access control in collaborative systems. As
highlighted in these works, scholars recently suggested solutions
to MPCs mainly following four different approaches.
      </p>
      <p>
        First of all, game-theoretic approaches model users as
rational entities and suggest the best option according to
some utility function users aim to maximise
        <xref ref-type="bibr" rid="ref16 ref21 ref25 ref27 ref29">(Squicciarini, Shehab, and Paci 2009; Such and Rovatsos 2016;
Rajtmajer et al. 2017)</xref>
        . While these proposals provided
elegant frameworks from a formal point of view and build upon
well-studied analytic tools, they may not work well when
used in practice
        <xref ref-type="bibr" rid="ref28">(Such and Criado 2018)</xref>
        . This is because
users’ behaviour does not seem perfectly rational in
practice, as assumed in these approaches
        <xref ref-type="bibr" rid="ref14 ref38">(Lampinen et al. 2011;
Wisniewski, Lipford, and Wilson 2012)</xref>
        , and very few other
factors are considered
        <xref ref-type="bibr" rid="ref21">(Rajtmajer et al. 2017)</xref>
        .
      </p>
      <p>
        The second type of approaches to support users when
solving MPCs base on recommendation engines, aiming to
suggest the best common sharing policy based on the
history of each user or of the platform
        <xref ref-type="bibr" rid="ref1 ref10 ref11 ref21 ref31">(Fogues et al. 2017a;
2017b)</xref>
        . However, it is not trivial to build a recommendation
model for MPCs with high accuracy and high flexibility, in
order to satisfy the user in a very dynamic environment; also
these systems are not transparent and self-explaining,
making it difficult for the user to understand the reason for which
they should take a particular decision.
      </p>
      <p>
        Another front of research proposes the use of AI
techniques developed from the multi-agent systems
community, from sets of norms
        <xref ref-type="bibr" rid="ref7">(Criado and Such 2015)</xref>
        , to
argumentation approaches
        <xref ref-type="bibr" rid="ref13 ref17 ref32">(Ko¨kciyan, Yaglikci, and Yolum
2017)</xref>
        or to negotiation frameworks
        <xref ref-type="bibr" rid="ref27 ref27 ref29 ref29">(Such and Criado 2016;
Such and Rovatsos 2016)</xref>
        . These approaches aim to support
users while deciding the most socially preferred sharing
action. Yet, the mechanisms may be difficult to comprehend
and/or too rigid, often demanding too much effort from the
user
        <xref ref-type="bibr" rid="ref1 ref28">(Such and Criado 2018; Acquisti et al. 2017)</xref>
        .
      </p>
      <p>
        The fourth and final type of approach uses data processing
(such as image processing)
        <xref ref-type="bibr" rid="ref12 ref35">(Ilia et al. 2015; Vishwamitra et
al. 2017)</xref>
        to tackle MPCs in social networks. User can
manipulate the images before sharing them online by, for
example, blurring the faces of the depicted people who do not
concede access to that item to third parties. In this case, the
main drawback is that, even without guaranteeing the
protection of the user’s privacy (someone can still be identified
by other details than the face), there is a significant loss of
sharing utility, as blurred pictures may not be as enjoyable.
      </p>
      <p>
        The first three types of approaches focus on
supporting the users involved in finding and agreement on a
common sharing policy for the data in different ways, while the
fourth type focuses on enforcing individual privacy
policies. Both approaches are complementary. For instance,
if there is an agreement possible between the users
involved, then this has the potential to be more satisfying
both in terms of privacy but also in terms of loss of
sharing utility, and empirical evidence tells us that many of the
MPCs could indeed be solved in practice
        <xref ref-type="bibr" rid="ref17 ref28 ref30 ref32">(Such et al. 2017;
Such and Criado 2018)</xref>
        . If, however, an agreement is not
found or is not possible, then something like blurring
pictures may provide a last resort baseline privacy.
      </p>
      <p>
        In this paper, we focus on supporting users to find an
agreement. The model we propose differs from the three
approaches mentioned above that also focus on supporting
users reaching an agreement, as it focuses on transparency
and on suggesting actions closer to users’ motivations and
values, both lacking in previous literature as detailed in the
paragraphs above. In particular, the model follows previous
works on usable privacy, which show the importance of
including in the models information to guide users towards
safer and better choices, without imposing a particular
decision
        <xref ref-type="bibr" rid="ref1">(Acquisti et al. 2017)</xref>
        . We hypothesise that a valid
contribution to the solution of MPCs is to include in the
model a self-explaining component, that makes it easier for
the user to understand and endorse the model’s
recommendation. The model produces suggestions along the
negotiation that promote or demote users’ values, like a Jiminy
Cricket from Pinocchio, to help users pick the actions that
are most aligned with their values.
      </p>
    </sec>
    <sec id="sec-3">
      <title>The Model</title>
      <p>Here we describe in detail the model to support the users
while reaching a collaborative decision on a common
sharing policy for a given piece of data that is co-owned by
them or that affects their privacy (e.g. a photo in which
multiple users are depicted). The model defines a negotiation
framework where each user is driven through the dialogue
by the values she considers important. The model takes as
input from every user involved in the decision (e.g. those
depicted in a photo) what would be their preferred sharing
policy for the item to be shared and their value order,
defining the relative importance each basic value has for them
(both - policy and value ordering - can be elicited with
minimal user intervention as detailed below). The model
supports all sides of the negotiation suggesting the action which
best suits each user’s preferences and value order, and it
provides feedback on how other possible options impact on the
promotion/demotion of those same values, leaving the user
the last word of the decision.</p>
      <sec id="sec-3-1">
        <title>Individual Sharing Policies</title>
        <p>
          We represent a collaborative platform (e.g. an OSN) where
agents are supposed to interact with each other and share
contents as a social graph G = (U; E; I), where the nodes
u 2 U are the users, the edges eij = (ui; uj ) 2 E are the
links between the users, and each edge has a weight
representing the closeness or intimacy of the relationship
between the two connected users; such intimacy i takes
values in f1; :::; Nig N, with Ni being the maximum, and
can be elicited by using predictive techniques as presented
in
          <xref ref-type="bibr" rid="ref34 ref8 ref9">(Fogue´s et al. 2014)</xref>
          .
        </p>
        <p>Content is shared in the platform according to sharing
policies, which define the criteria users must satisfy in
order to access such content.</p>
        <p>Definition 1. A sharing policy is a tuple p = hd; ii 2 P,
where d 2 [0; Nd] represents the maximum allowed distance
a user must be from the owner of the content, meant as the
length of the shortest path of the social graph connecting
the two users, and i 2 [0; Ni] represents the minimum
required intimacy over each edge of the path connecting the
two users.</p>
        <p>
          Note that the policy definition used in this paper could
be translated to and back from the usual group-based access
control policies of social media sites like Facebook
          <xref ref-type="bibr" rid="ref27 ref29">(Such
and Criado 2016)</xref>
          . Also, note that we assume that individual
privacy preferences for each item, i.e. the sharing policy p
each of the users involved would prefer if they were to
decide about the item alone, are provided directly by the user
or they are elicited, to minimise user effort, following
datadriven AI techniques like machine learning shown to work
very well to elicit individual privacy preferences in social
media while minimising user effort such as
          <xref ref-type="bibr" rid="ref17 ref24 ref30 ref32">(Squicciarini et
al. 2011; Misra and Such 2017)</xref>
          , or derived from suitable
defaults based on approaches like
          <xref ref-type="bibr" rid="ref37 ref7">(Watson, Lipford, and
Besmer 2015)</xref>
          .
        </p>
      </sec>
      <sec id="sec-3-2">
        <title>Schwartz Basic Values</title>
        <p>
          We base on the theory of basic values by Schwartz
          <xref ref-type="bibr" rid="ref22">(Schwartz 2010)</xref>
          to model human values, as this is the most
well-known and established theory of values with strong
empirical evidence backing the theory, validated in many
studies and over different countries
          <xref ref-type="bibr" rid="ref6">(Bilsky, Janik, and Schwartz
2011)</xref>
          , and which has been successfully applied in
various contexts, including environmentalism
          <xref ref-type="bibr" rid="ref26">(Stern, Dietz, and
Guagnano 1998)</xref>
          , recruitment
          <xref ref-type="bibr" rid="ref20">(Patterson et al. 2016)</xref>
          ,
consumers habits
          <xref ref-type="bibr" rid="ref27 ref29 ref33">(Thøgersen, Zhou, and Huang 2016)</xref>
          ,
healthcare
          <xref ref-type="bibr" rid="ref18">(Moyo et al. 2016)</xref>
          , and many others.
        </p>
        <p>
          Values are defined by Schwartz as socially desirable
concepts that allow humans to interact between themselves,
representing mental goals and the way used to describe and
communicate such goals
          <xref ref-type="bibr" rid="ref23">(Schwartz 2012)</xref>
          . People take daily
decisions according to the values they believe into. Values
compete with each other, but the individual realises the
dissonance and decides what to do by giving priority to some
values over the others.
        </p>
        <p>In our context, we identify five relevant categories of basic
values from Schwartz’ theory:
self-direction (sd): the user is open-minded and ready to
change the negotiating strategy during the decision
making process to suggest new solutions;
power (po): the user holds her initial idea, giving no space
to accommodate the others’ preferences;
security (se): the user prefers the safer option, in this case
the most restrictive one in terms of publicity;
conformity, tradition (tr): the user’s choice is highly
influenced by the society’s expectations;
benevolence, universalism (be): the user is willing to
consider any proposal that is coming from a close friend
(benevolence) or anyone (universalism).</p>
        <p>We formalise the relative importance of the values for
each user as follows.
Definition 2. A value order is a particular order o 2 O
over the values V = fbe; po; se; sd; trg that determines the
relative relevance the user believes each value has on her
behaviour, where O is the space of all the possible total or
partial orders over V.</p>
        <p>Example: A user has a total order over her values o :
po se sd tr be, meaning that she considers power
as the most important value to guide her behaviour, followed
by security as the second most influential and so on.</p>
        <p>
          Note that, in practice, the importance that each user
attributes to each of these values can be elicited using the
Schwartz Value Survey or the Portrait Values
Questionnaire validated by Schwartz
          <xref ref-type="bibr" rid="ref23">(Schwartz 2012)</xref>
          . Also note that,
while the example shows a total order, our model would also
work with partial orders, e.g., users having some preferences
of values over the others but not for all values.
        </p>
      </sec>
      <sec id="sec-3-3">
        <title>Negotiation process</title>
        <p>
          The communication between the users involved follows a
number of steps (or negotiation rounds) until a decision
is agreed. For simplicity and without loss of generality,
we will consider in this paper just two users and an
alternated proposals negotiation framework
          <xref ref-type="bibr" rid="ref8">(Fatima, Kraus, and
Wooldridge 2014)</xref>
          , depicted in the Figure 1. The uploader
A starts the dialogue offering a certain policy p to the
coowner B. B evaluates the received offer: if B accepts, then
the negotiation is over and the content is shared with
policy p ; if B is not satisfied by the offer, then B can make a
counter-proposal, which is evaluated by A and so on.
However, there might be no possibility of reaching an agreement,
for instance if both parties keep offering the same policy
without trying to accommodate each other. In this case, the
negotiation is considered as failed and no content is shared.
        </p>
        <p>
          In order to support the user while taking decisions during
the negotiating process, at every iteration the model suggests
to each user the action that is the most coherent one with the
ordering of their values, as detailed in the next section in
Definition 6. As it is important to keep the user in the loop
and in full control if the user desires so, as otherwise some
users will feel out of control of their privacy
          <xref ref-type="bibr" rid="ref34">(Vihavainen et
al. 2014)</xref>
          , the model can also produce other options together
with how these other options promote/demote particular
values. After a user suggests a policy, the other user has to
decide if she accepts it. The condition for accepting depends
on the distance between the received policy proposal and the
one which would be suggested by the other user in case of
rejection. Again, this may be left as a default distance value
or to be decided by the user.
        </p>
        <p>Definition 3. The distance between two policies p and
p is defined as the Manhattan distance:
= jd
d j + ji
i j;
where p = hd ; i i and p = hd ; i i are respectively the
preferred policy by users A and B.</p>
        <p>If such distance is within a reasonable range, for instance
it is equal to 0 so that the policies are the same or equivalent,
then the offer is accepted and the negotiation ends.
Otherwise, the other user communicates her counter-proposal and
the dialogue proceeds until either the convergence is reached
or until it is recognised as impossible, i.e. when both users
have tried out all their strategies and cannot help but keep
suggesting the same policies. If an agreement is found, then
the item in discussion is shared according to the last offered
policy. Otherwise, the content remains private.</p>
      </sec>
      <sec id="sec-3-4">
        <title>Generation of a new policy proposal</title>
        <p>We now introduce some definitions that help us understand
the process through which the model suggests to the user
the policy most coherent with her values at each negotiation
round.</p>
        <p>Definition 4. A value-function fv : P3 ! P3 defines the
cumulative effect of a value on the policy to be suggested. It
takes as inputs the preferred policies p ; p of the two users,
and pv, a policy which memorises the outcome of the other
value-functions previously combined. The codomain of fv
is a subset of its domain: i.e. given three policies p ; p ; pv
and defining
p1 =hmin(d ; d ; dv); max(i ; i ; iv)i = hd1; i1i
p2 =hmax(d ; d ; dv); min(i ; i ; iv)i = hd2; i2i
as the tuples having respectively the most and the least
restrictive components over p ; p ; pv, then
fv(p ; p ; pv) 2 ([d1; d2]
[i2; i1])3:
(1)</p>
        <p>
          There are four fv in the space of the value-functions
F = ffbe; fpo; fse; ftrg representing respectively the
influence of benevolence, power, security and tradition on the
policy offer. A possible instance of the fv functions is shown
in Table 1. For the case of the self-direction value, we do not
model it as a value function but instead we model it as a
different function that changes the particular value ordering,
as whenever the user believes in self-direction, the
importance of being creative and open-minded while researching
a solution is what matters
          <xref ref-type="bibr" rid="ref22">(Schwartz 2010)</xref>
          .
        </p>
        <p>fv
fpo(p ; p ; pv) = (p ; p ; avg(p ; pv))
fbe(p ; p ; pv) = (p ; p ; avg(p ; pv))
fse(p ; p ; pv) = (p ; p ; avg(hminp ;p d; maxp ;p ii; pv))
ftr(p ; p ; pv) = (p ; p ; avg(p ; pv))
Definition 5. The sd-function fsd : O ! O defines the
influence of the value self-direction over the entire user’s
strategy. Considering the order provided by the user, it returns
another order where the value self-direction is removed and
the two following values, if any, are swapped at every other
iteration of the negotiation.</p>
        <p>The sd-function allows the user to be flexible in her
negotiation process by employing eventually two strategies
alternatively, i.e. two different orders over benevolence, power,
security and tradition.</p>
        <p>We are finally ready to introduce the crucial part of the
model, that is the generation of a new proposal. At each step
of the negotiation after the first one, a user receives an offer
that needs to be evaluated. To do so, the system generates
for the user what should be her next suggestion, given her
preferred policy and order over values.</p>
        <p>Definition 6. A proposal generator g : P3 ! P is a
function which provides the policy that should be suggested
according to the particular value ordering and the initial policy
preferences of the two users. In particular, it is a
composition of value-functions fi 2 F , where the order of the
composition1 is given by the order over values o 2 O, and the
projection operator 3, which selects only the third policy
from the last obtained tuple.</p>
        <p>The proposal generator g computes for the user A what
the next policy proposal p~ = g(p ; p ; o ) should be, so
that it is the policy most consistent with the user’s A
order over values and the initial preferred sharing policies (an
equivalent optimal policy is computed also for the user B,
whenever it is her time to evaluate an offer from A). Then,
according to the distance introduced in Definition 3, the user
decides whether to accept the other user’s proposal or to
make a counter-offer, in which case the system suggests the
last generated policy p~.</p>
      </sec>
      <sec id="sec-3-5">
        <title>Example Negotiation</title>
        <p>Consider a situation where users A and B discuss about
sharing some content on an online collaborative platform.</p>
        <p>1In case the value order is a total order, then the composition
order of the value functions is trivial (the Example Negotiation
section). If the value order is a partial order, different solutions can be
applied, for instance picking randomly only one of the equivalent
value-functions.</p>
        <p>Their preferred policies are respectively p = h5; 8i and
p = h1; 1i. According to the sensitivity of the content, in
general people would suggest the policy p = h2; 6i. A and
B also provide to the system their order over the values:
se
be
be
tr
tr
se
po
sd
sd
po:
We consider the instances of the value-functions listed in
Table 1, and that at the beginning of each step pv = null. Here,
both users have a single strategy, because self-direction is in
the last or second-last position in the order and therefore it
has no values to swap. In this situation, the condition to
accept an offer is to reach the same common policy, so that the
distance is = 0 from the newly generated one.</p>
        <p>At t = 0, A suggests the policy p0 = p = h5; 8i.</p>
        <p>At t = 1, B decides whether to accept or reject the
offer. To do this, the system computes the best (according to
B’s values) policy that B would eventually counter-offer
and then, if this coincides with p0, the system would
suggest to accept:
g(p ; p ; o ) = 3
fbe ftr</p>
        <p>
          fse fpo(p ; p ; null)
= 3
= 3
= 3
fbe(ftr(fse(h5; 8i; h1; 1i; h1; 1i)))
fbe(ftr(h5; 8i; h1; 1i; h1; 4i))
fbe(h5; 8i; h1; 1i; h1; 5i)
= 3(h5; 8i; h1; 1i; h3; 7i)
=h3; 7i:
Since = j5 3j + j8 7j &gt; 0, the system suggests B to
reject the offer and to propose p1 = h3; 7i. However, at every
iteration, the decision about following the system’s
suggestion is left to the user. Future studies should look into how
to convey to the users some interpretation and comments on
each of the possible actions they might take
          <xref ref-type="bibr" rid="ref1">(Acquisti et al.
2017)</xref>
          , for instance which values might be promoted or
demoted by suggesting another policy. For simplicity here we
are describing a case when each user always performs the
action which is suggested by the system as the most
coherent one given her own values. So, B rejects p0 and offers
p1 = h3; 7i.
        </p>
        <p>At t = 2, it’s A’s time to evaluate B’s offer:
g(p ; p1; o ) = 3
fse
fbe ftr</p>
        <p>fpo(p ; p1; null)
= 3
= 3
= 3
fse(fbe(ftr(h5; 8i; h3; 7i; h5; 8i)))
fse(fbe(h5; 8i; h3; 7i; h4; 7i))
fse(h5; 8i; h3; 7i; h3; 7i)
= 3(h5; 8i; h3; 7i; h3; 8i)
=h3; 8i:
Since = j3 3j + j8 7j &gt; 0, A rejects the offer and
proposes p2 = h3; 8i.</p>
        <p>At t = 3, B reasons about the last offer received from A:
g(p2; p1; o ) = 3
fbe ftr
fse</p>
        <p>fpo(p2; p1; null)
= 3
= 3
= 3
fbe(ftr(fse(h3; 8i; h3; 7i; h3; 7i)))
fbe(ftr(h3; 8i; h3; 7i; h3; 7i))
fbe(h3; 8i; h3; 7i; h3; 7i)
= 3(h3; 8i; h3; 7i; h3; 8i)
=h3; 8i:
Since B obtains the same policy that is last offered by A, B
accepts and the content is shared with policy p = h3; 8i.</p>
      </sec>
    </sec>
    <sec id="sec-4">
      <title>Properties of the model</title>
      <p>Given the practical problem we aim to solve, it is crucial for
the model to present some properties that allow its
implementation on real systems, like termination in a finite time,
correctness and completeness.</p>
      <p>Lemma 1. Termination On the assumption that neither
party withdraws, in a finite time, the offers pt suggested by
the system always converge towards an agreement p~
d(pt; p~) ! 0 for t ! N &lt; +1
(2)
or the system recognises the impossibility of reaching an
agreement.</p>
      <p>Proof. During the negotiation process, a user can either
maintain her position or accommodate the other user’s
preference. Let us consider each case separately:
(a) Both users want to accommodate each other: let us
hypothesise, by absurd, that (2) is false; this means that the
distance between the new suggestion and the final
agreement may increase at each iteration or that the
convergence may happen in an infinite number of iterations. By
the definitions of the functions fv, the output of each fv is
always within the range defined by the most and the least
restrictive tuples of each iteration (see Equation (1)). If the
users are both trying to accommodate each other’s
preference, it means that the new suggestion is a tuple whose at
least one element is internal to the domain; i.e. the domain
of fv(p ; p ; pv) becomes one of the following:
[d1; d2)
[d1; d2)
(d1; d2)
(i2; i1]
[i2; i1)
(i2; i1):
(d1; d2]
(d1; d2]
(i2; i1]
[i2; i1)
Noting that these are all subsets of N2, it follows that the
width of the domain of the new suggestion decreases at
every iteration. Therefore, the distance between the new
suggestion and the final deal can only decrease, as they
both are elements of the domain; this contradicts our
initial hypothesis. Also, given the fact that the domain is a
finite and bounded subset of N2, the convergence happens
in a finite number of iterations. Therefore, (2) is valid.
(b) One user wants to accommodate, the other user holds her
position: the reasoning is similar to the previous case, but
now the contraction of the domain happens only at every
other iteration, i.e. whenever a user makes an
accommodating offer. In fact, when a user sticks to her preference,
she keeps offering a tuple whose elements are on the
border of the domain. Eventually with a slower speed than
in the previous case, the domain does get contracted and,
given its finite dimension and its boundary, it converges
in a finite time. So, (2) is valid.
(c) At some point, both users start holding their positions: if
both users stick to their preferences, it means that both of
them keep suggesting policies whose elements are on the
border of the domain
fv(p ; p ; p ) 2 [d1; d2]
[i2; i1] 2 N2;
therefore, the domain cannot contract. The system
interrupts the negotiation whenever both users have tried all
their strategies after receiving the same inputs. Since
every user has at most two strategies, this happens at latest
at the 5th iteration of the algorithm with no changes in the
offers: that is, the termination of the algorithm is realised
in a finite time.</p>
      <p>Lemma 2. Correctness Assuming that the users always
follow the system’s suggestions, the outcome of the negotiation
is consistent with their initial preferences and orders over the
basic value.</p>
      <p>Proof. This can be proved by contradiction. Let us assume
that the outcome of the negotiation is not consistent with
the users’ initial preferences over the sharing policies and
the values, even though both users always followed the
system’s suggestions. This implies that, in at least one step of
the negotiation process, the system provided a policy
recommendation which was not consistent with the users’ inputs.
A new policy proposal is defined from the function g (see
Definition 6) as the composition of, excluding the final
projection, only fv functions (see Equation (1)). If a new
suggestion given by the function g is not consistent, it means
that either (i) the composition order or (ii) the fv functions
are not consistent with the inputs. However, (i) the order for
composing the fv functions is defined exclusively by the
order over values provided by the users: this assures that
different relevance is given to different fv functions according
to the priority that the user assigns to the values that each fv
function represents. Therefore, the composition order is by
definition consistent with the user’s preference. On the other
hand, (ii) the fv functions are defined in such a way that
they reflect the interpretation of each value in the
negotiation context given the initial policy preferences; for instance,
in Table 1 fpo reflects power and is influenced by whatever
the output of the negotiation is up to that point (pv) and by
the last policy preference of the user (for the user A, p in
the first step and then the last policy proposed by A). So,
by definition, the value-functions are consistent with the
initial preferences of the users. In both (i) and (ii) we reached
a contradiction, therefore we can say that, given a coherent
behaviour from the users’ side, every step of the
negotiation must be consistent with the initial preferences of the
users. We know that every outcome of the model is reached
through a sequence of such consistent steps; therefore every
outcome whether it is a deal or no deal, must be consistent
with the initial preferences of the users.</p>
      <p>Lemma 3. Completeness An optimal agreement is reached
by a sequence of optimal negotiation steps; the outcome of
a single step is optimal if it is coherent with the user’s initial
preferences. If an optimal agreement exists, then the system
is able to find it.</p>
      <p>Proof. An optimal agreement is the outcome of a chain of
actions that are coherent with the user’s initial preferences,
i.e. when the user follows the system’s suggestion instead of
acting impulsively. Whenever the chain of coherent events
is altered by an impulsive choice, it becomes impossible
to explain or predict this outcome using the system’s rules
without introducing an inconsistency. Therefore the system
is complete.</p>
    </sec>
    <sec id="sec-5">
      <title>Conclusion</title>
      <p>
        In the past years, an increasing attention has been paid to
MPCs on online collaborative platforms, especially on
social networks. Reparative solutions are proved not to be
efficient and many attempts have been performed by scholars
to tackle this problem, while considering collaborative
decision making strategies. However, the majority of the
suggested solutions fail to provide simple and understandable
reasons about the generation of their suggestions, sometimes
hindering the user’s endorsement of the recommended
policy provided by the model. Following previous studies on
usable security
        <xref ref-type="bibr" rid="ref1">(Acquisti et al. 2017)</xref>
        , we hypothesise that the
transparency and the interpretability of a model can play a
crucial role in solving MPCs. On this basis, we built a model
to support the users while interacting with each other in
order to collaboratively define access control based on values.
Following the model’s suggestions at every negotiation step,
each user is recommended to act coherently with her values,
and when an agreement is achieved, the agreement is
guaranteed to be the most consistent with their initially preferred
sharing policies and her values.
      </p>
      <p>
        Despite the rigorous and formally proven definition of
the model in this paper, this line of research is still
workin-progress. There are some particular directions we would
like to follow. First, we would like to validate the model here
presented and refine it with empirical evidence coming from
user studies. Particularly, we would like to test the
hypothesis that users are able to understand better the support both
during negotiations and after an outcome is agreed and that a
value-based approach leads to better user satisfaction when
resolving MPCs. Before we can perform the user study, it is
crucial that we define the best way to maximise the usability
of any interfaces based on our model: this includes studying
the best trade-off between the autonomy of the system and
the user’s effort, and identifying the most suitable
presentation of actions promoting/demoting values for each action,
which we will do following evidence from privacy nudges
and notifications
        <xref ref-type="bibr" rid="ref1">(Acquisti et al. 2017)</xref>
        . Secondly, the aim
of this model is to capture a single collaborative decision
processes. Given that interactions over a OSN have an
evolutionary nature, it seems interesting for us to analyse the
influence over time that the outcomes of particular
negotiations have on the overall users’ behaviour, i.e. if it is possible
to identify an historical component of the decision making
process defined by the previous experiences. Also, the
formal syntax and semantics for the negotiation protocol need
to be defined
        <xref ref-type="bibr" rid="ref16">(McBurney and Parsons 2009)</xref>
        .
      </p>
    </sec>
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