Josang trust algebra

Audun Jøsang defined a complete trust algebra based on subjective logic. Subjective logic is a belief reasoning calculus that is compatible with, and an extension of probabilistic logic. It can for example be used for modeling trust networks, and for analysing beliefs in Bayesian networks. In general, subjective logic is suitable for modeling and analysing situations involving uncertainty and incomplete knowledge.

The basic idea is that uncertainty is added formally in the modeling. While in general we can formalize "trust+distrust=1", in subjective logic application to trust, we have "trust+distrust+uncertainty=1". For example, when

The opinion of A about B (trust statement) is hence a point contained into a triangle which sums to 1. For example, when A does not know B and have no opinion on her, the trust statement is a point toward the top of the triangle near the Uncertainty vertex (for example, it can be (0.1,0,0.9) meaning that A does not know if she can trust B (0.9) but is willing to give a small initial trust as well (0.1). After A comes to know B based on few real interactions, A can decide for example that she trusts B (0.7) but she is still not totally sure about this (uncertainty=0.2) and so to update the trust statement into a somehow more positive one, i.e a point close to the bottom right Trust vertex (such as (0.7,0.1,0.2)).

The triangle could also be visualized as a color, mixing white (uncertainty) with green (trust) and red (no trust).

See demos. There is also a demo for trust network propagation.

Formula to propagate trust: $$\omega^{A}_{D} = (\omega^{A}_{B}\otimes \omega^{B}_{D}) \oplus (\omega^{A}_{C}\otimes \omega^{C}_{D})\,\!$$