Reasoning about the impacts of information sharing

Abstract

Shared information can benefit an agent, allowing others to aid it in its goals. However, such information can also harm, for example when malicious agents are aware of these goals, and can then thereby subvert the goal-maker’s plans. In this paper we describe a decision process framework allowing an agent to decide what information it should reveal to its neighbours within a communication network in order to maximise its utility. We assume that these neighbours can pass information onto others within the network. The inferences made by agents receiving the messages can have a positive or negative impact on the information providing agent, and our decision process seeks to assess how a message should be modified in order to be most beneficial to the information producer. Our decision process is based on the provider’s subjective beliefs about others in the system, and therefore makes extensive use of the notion of trust with regards to the likelihood that a message will be passed on by the receiver, and the likelihood that an agent will use the information against the provider. Our core contributions are therefore the construction of a model of information propagation; the description of the agent’s decision procedure; and an analysis of some of its properties.

Appendix : The Case of Continuous Random Variables

By utilising Definitions 8 and 9 we can describe the impact of disclosing a message to the consumers on the producer a g ₀.

Proposition 1

Given a FCA ${\langle {\mathcal {A}}, {\mathcal {C}}, \mathcal {M}, m \rangle }$;a consumer ${ {\mathit {ag}_{q}}} \in {\mathcal {A}}$;and the equivalent degree of disclosure x _0,q of the producer over ag _q . Let y _q be the information inferred by ag _q according to the r.v. I _q (x _0,q ) (with probability $\approx f_{I_{q}}(y_{q};{x}_{0,q}) \, dy_{q}$ ). Then, assuming that the impact z _q is independent of the degree of disclosure distribution x _0,q given the inferred information y _q , ag ₀ expects an impact z _q described by the r.v. Z _q (x _0,q ) with density:

$$f_{Z_{q}}(z_{q} ; x_{0,q}) = {{\int}_{0}^{1}} f_{Z_{q}} (z_{q} ; y_{q}) \, f_{I_{q}} (y_{q} ; {x}_{0,q} ) \, d y_{q} . $$

Proof

$$\begin{array}{@{}rcl@{}} F_{Z_{q}}(z_{q} ; {x}_{0,q}) &=& \Pr \{ Z_{q} \leq z_{q} | {x}_{0,q} \} \\ &=& {\int_{0}^{1}} \Pr \{ Z_{q} \leq z_{q}, I_{q} = y_{q} | {x}_{0,q} \} \, dy_{q} \\ &=& {\int_{0}^{1}} \Pr \{ Z_{q} \leq z_{q} | I_{q} = y_{q} , {x}_{0,q} \} f_{I_{q}}(y_{q};{x}_{0,q}) \, dy_{q} \\ &=& {\int_{0}^{1}} F_{Z_{q}} (z_{q} ; y_{q}) \, f_{I_{q}} (y_{q} ; {x}_{0,q} ) \, d y_{q} , \end{array} $$

The density function is easily derived from the distribution $F_{Z_{q}}(z_{q} ; {{x_{0,q}}})$ since $f_{Z_{q}}(z_{q} ; {x}_{0,q}) = \frac { d }{d z_{q}}F_{Z_{q}} (z_{q} ; {x}_{0,q})$.

Moreover, any time we need a single value characterisation of a distribution, we can exploit the same idea of descriptors of a random variable, by introducing descriptors for trust and risk.

Definition 13

Let h(⋅) be a function defined on [0,1], and y∈[0,1] be a level of inference. We define

$$ t_{h}^{Z_{q}} (x) = {{\int}_{0}^{1}} h(w) \, f_{Z_{q}} (w;y) \, d w , $$

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to be the y-trust descriptor induced by h(⋅).

We can do the same to obtain a impact descriptor:

Definition 14

Let h(⋅) be a function defined on [0,1], and x∈[0,1] be a level of disclosure.We define

$$ t_{h}^{Z_{q}} (x) = {{\int}_{0}^{1}} h(w) \, f_{Z_{q}} (w;x) \, d w , $$

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to be the x-impact descriptor induced by h(⋅).

Typical h(⋅) include the moment generating functions, such as h(k) = k,k ², etc., and entropy $h(k)= - ln \bigl (\allowbreak f_{K} (k) \bigr )$ for the density of some r.v. K. In the following we use the expectation as the risk descriptor, leaving consideration of other possible functions for future work.

Finally, let us illustrate two notable properties of our model. The first one is with regards to the case where a consumer can derive the full original message, which, unsurprisingly, leads to the worst case impact.

Proposition 2

When a consumer is capable of gaining maximum knowledge, then f _I (y;x)=δ(y−1), where δ(⋅) is the Dirac delta function, and $F_{Z} (z ; x ) = F_{Z} (z) \triangleq F_{Z} (z;1)$, i.e., the risk coincides with the 1-trust (Definition 9).

Proof

By the definition of the inference r.v. I(x), when a g _x is believed to gain maximum knowledge then the density f _I (y;x) carries all its weight at the point y=1 for all x. Hence, f _I (y;x) = δ(y−1) and it follows from the definition of the Dirac delta function, see also Prop. 1

$$\begin{array}{@{}rcl@{}} F_{Z} ( z ; x ) &=& {\int_{0}^{1}} F_{Z} (z ; y) \, f_{I} (y ; x ) \, d y \\ &=& {\int_{0}^{1}} F_{Z} (z ; y) \, \delta (y-1) \, d y = F_{Z} (z;1) . \end{array} $$

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The second property pertains to the case where agent a g ₀ shares information with more than one consumer. Such situations are typically non-homogeneous as the trust and impact levels with regards to each consumer are different. Clearly, it is beneficial to identify conditions where these impacts balance (and, hence, indicate crossover thresholds) across the multiple agents.

For two agents a g ₁, a g ₂ having corresponding inference and behavioural trust distributions $F_{I_{j}} (y;x)$ and $F_{Z_{j}} (z;y)$, j∈{1,2}, for the shared information to have similar impact, x ₁ and x ₂ should be selected, such that the following holds.

$$\begin{array}{@{}rcl@{}} F_{Z_{1}}(z ; x_{1}) & =& F_{Z_{2}}(z ; x_{2}) \Leftrightarrow \\ {{\int}_{0}^{1}} F_{Z_{1}} (z;y) \, f_{I_{1}} (y;x_{1}) \, d y & =& {{\int}_{0}^{1}} F_{Z_{2}}(z;y) \, f_{I_{2}} (y;x_{2}) \, d y . \end{array} $$

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Note that the above relationship implies the r.v.s Z ₁ and Z ₂ are drawn from the same distribution. Such a requirement is typically unrealistic. Therefore in general one may want to consider equalities on the average, such as, finding x ₁ and x ₂ satisfying the following for appropriate g(⋅) functions.

$$ \mathbb{E}\{ g (Z_{1} (x_{1}) ) \} = \mathbb{E}\{ g (Z_{2} (x_{2}) ) \} , $$

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Proposition 3

Given that g(z) = z, in order to attain the same level of impact when ag ₀ shares information with ag ₁, ag ₂, the degrees of disclosure x ₁ and x ₂ for ag ₁ ,ag ₂ respectively must satisfy the following.

$$ \mathbb{E}_{I_{1}} \bigl \{ \mathbb{E} \{ Z_{1} (x_{1}) | I_{1} \} \bigr \} = \mathbb{E}_{I_{2}} \bigl \{ \mathbb{E} \{ Z_{2} (x_{2}) | I_{2} \} \bigr \} . $$

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Proof

The case where g(z) = z corresponds to the regular averaging operator, and Eq. 13 becomes:

$$\begin{array}{@{}rcl@{}} &&{\int_{0}^{1}} {\int_{0}^{1}} z f_{Z_{1}} (z ; y) \, f_{I_{1}} (y ; x_{1} ) \, d y \, dz \\ &&\kern1.5pc= {\int_{0}^{1}} {\int_{0}^{1}} z f_{Z_{2}} (z ; y) \, f_{I_{2}} (y ; x_{2} ) \, d y \, dz \\ &&\Leftrightarrow \\ &&{\int_{0}^{1}} f_{I_{1}} (y ; x_{1} ) \biggl [ {\int_{0}^{1}} z f_{Z_{1}} (z ; y) \, d z \biggr ] \, dy \\ &&\kern1.5pc= {\int_{0}^{1}} f_{I_{2}} (y ; x_{2} ) \biggl [ {\int_{0}^{1}} z f_{Z_{2}} (z ; y) \, d z \biggr ] \, dy \\ && \Leftrightarrow \\ && \mathbb{E}_{I_{1}} \bigl \{ \mathbb{E} \{ Z_{1} (x_{1}) | I_{1} \} \bigr \} \\ && \kern1.5pc= \mathbb{E}_{I_{2}} \bigl \{ \mathbb{E} \{ Z_{2} (x_{2}) | I_{2} \} \bigr \} . \end{array} $$