Concepts for decision making under severe uncertainty with partial ordinal and partial cardinal preferences

Abstract

We introduce three different approaches for decision making under uncertainty if (I) there is only partial (both cardinally and ordinally scaled) information on an agent's preferences and (II) the uncertainty about the states of nature is described by a credal set (or some other imprecise probabilistic model). Particularly, situation (I) is modeled by a pair of binary relations, one specifying the partial rank order of the alternatives and the other modeling partial information on the strength of preference. Our first approach relies on decision criteria constructing complete rankings of the available acts that are based on generalized expectation intervals. Subsequently, we introduce different concepts of global admissibility that construct partial orders between the available acts by comparing them all simultaneously. Finally, we define criteria induced by suitable binary relations on the set of acts and, therefore, can be understood as concepts of local admissibility. For certain criteria, we provide linear programming based algorithms for checking optimality/admissibility of acts. Additionally, the paper includes a discussion of a prototypical situation by means of a toy example.

Normal cones corresponding to credal sets of lower probabilities

“…Assuming the above convention, the credal set of a coherent lower prevision M(P ) is the set of vectors p satisfying constraints (17) and (19).…”

“…Normal cones corresponding to credal sets of lower previsions will be equivalently denoted by N (P , P ) = N (M(P ), P ) = {f : P (f ) = P (f )} = cone(F(P )), where F(P ) = {f ∈ F : P (f ) = P (f )}. In addition to inequality constraints, credal sets satisfy the equality constraint (19). This implies that every F(P ) contains the constant gamble I X .…”

“…In recent years, methods of imprecise probabilities [3,4,37] have been applied to various areas of probabilistic modelling, such as stochastic processes [10,52], game theory [26,30], reliability theory [8,21,31,49,59], decision theory [19,28,46], financial risk theory [36,51], computer science [1,38,48,50], copulas [14,32,33,34,60] and others.…”

Relaxing Time Windows by Partial Orders in Routing Problems With Stacking Constraints

“…In VRPs considered here, a partial order relation refers to a strict precedence order existed between a pair of tasks. Based on the partial order set theory [11]- [13], given a task set T , '' '' represents the partial order relation in T . Partial order relation has the following characters: reflexive, for any x ∈ T , x…”

Information efficient learning of complexly structured preferences: Elicitation procedures and their application to decision making under uncertainty

“…The references that seem most relevant in our context are Danielson and Ekenberg (1998), Danielson et al (2003) and Troffaes and Sahlin (2017) as these, similar to us, also allow for partial preferences and investigate connections to decision making under imprecise probabilities. For a discussion of relations and differences of these and our approach see Jansen et al (2018). For a recent discussion of the topic from a rather philosophical point of view see, e.g., Baccelli and Mongin (2016).…”