A Theory of Three-way Decisions

Three-way decisions are thinking in threes. In contrast to dichotomous thinking in terms of two options, three-way decisions introduce a third option. We move from true/false, black/white, yes/no etc. into true/unsure/false, black/grey/white, yes/maybe/no etc. The third option provides the necessary flexibility and universality of thinking in threes.

A theory of three-way decisions (3WD) may be interpreted within a trisecting-acting-outcome (TAO) framework (see reference [1] below). With respect to trisecting, we divide a whole into three parts. With respect to acting, we design most effective strategies for processing the three parts. With respect to outcome, we evaluate the effectiveness of the combined outcome of trisection and acting. In a set-theoretic model of three-way decisions, we divide a universal set into three pair-wise disjoint regions known as a trisection or a weak tri-partition. We design most effective strategies for processing the three regions. The identification and explicit investigation of different strategies for different regions are a distinguishing feature of three-way decisions.

A specific model of three-way decisions may be formulated based on the notions of acceptance, rejection and noncommitment. It is an extension of the commonly used binary-decision model with an added third option. Three-way decisions play a key role in everyday decision-making and have been widely used in many fields and disciplines.

The notion of three-way decisions was originally introduced by the needs to explain the three regions of probabilistic rough sets (see reference [3] below). One can construct rules for acceptance from the positive region and rules for rejection from the negative region. When neither a decision of an acceptance nor a decision of rejection can be made, a third option of noncommitment is chosen. Recent studies show that rough set theory is only one of possible ways to construct three regions. A more general theory of three-way decisions has been proposed (see reference [2] below), embracing ideas from rough sets, interval sets, shadowed sets, three-way approximations of fuzzy sets, orthopairs, square of oppositions, and others.

[1] Y.Y. Yao, Three-way decision and granular computing, International Journal of Approximate Reasoning 103 (2018) 107-123.
[2] Y.Y. Yao, An outline of a theory of three-way decisions, in: J.T. Yao, Y. Yang, R. Slowinski, S. Greco, H. Li, S. Mitra, L. Polkowski (Eds.), Rough Sets and Current Trends in Computing, LNCS 7413, Springer, Heidelberg, 2012, pp. 1-17.
[3] Y.Y. Yao, Three-way decisions: an interpretation of rules in rough set theory, in: P. Wen, Y.F. Li, L. Polkowski, Y.Y. Yao, S. Tsumoto, G.Y. Wang (Eds.), Rough Sets and Knowledge Technology, LNCS 5589, Springer, Heidelberg, 2009, pp. 642-649.