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On the hierarchical nature of partial preferences over lotteries

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Abstract

In this work we consider preference relations that might not be total. Partial preferences may be helpful to represent those situations where, due to lack of information or vacillating desires, the decision maker would like to maintain different options “alive” and defer the final decision. In particular, we show that, when totality is relaxed, different axiomatizations of classical Decision Theory are no longer equivalent but form a hierarchy where some of them are more restrictive than others. We compare such axiomatizations with respect to theoretical aspects—such as their ability to propagate comparability/incomparability over lotteries and the induced topology—and to different preference elicitation methodologies that are applicable in concrete domains. We also provide a polynomial-time procedure based on the bipartite matching problem to determine whether one lottery is preferred to another.

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Notes

  1. Together with Newell, he won the Turing Award in 1975 for his contributions on “artificial intelligence, the psychology of human cognition, and list processing” (http://amturing.acm.org/award_winners/simon_1031467.cfm).

  2. In Greek mythology, Eteocles and Polynices, sons of Oedipus, contended for the city of Thebes and killed each other in battle.

  3. A preorder is a reflexive and transitive relation, clearly Axiom 1 implies reflexivity.

  4. To the best of our knowledge, such an extensive analysis of axioms’ interdependencies has not been shown elsewhere.

  5. By “linearly closed” the authors means that the intersection of D with any line is a closed set. This in particular implies that D is closed.

  6. We refer to [1] for a concrete application domain where a user/agent interface provides a set of statements \(\mathsf {S}\) over service-level agreements.

  7. It is an open question whether Theorem 20 holds for \(\text {PDT}_{4/5}\).

  8. Instantiating the relations \(=_c\) and \(<_c\) is not more complex than calculating the transitive closure of a relation, hence it can be done in \(O(\vert \mathcal {A} \vert ^3)\). Then, on the one hand, checking \(f\sim _c g\) means to verify that, for all equivalent classes \([a]]\) induced by \(=_c\), \(f([[a]])=g([[a]])\), which can be done in \(O(\vert \mathcal {A} \vert )\). On the other hand, checking \(f\prec _c g\) means to verify that \(f\twoheadrightarrow _c g\), by Theorem 6 this is essentially equivalent to a bipartite-matching problem that can be solved again in \(O(\vert \mathcal {A} \vert ^3)\) [47].

  9. Here, \(\delta \) is interpreted as a constant function and hence \((Y + \delta )(\omega )=Y(\omega )+\delta \).

  10. Note that this can happen even if the user’s statements are a total order of \(\mathcal {A} \). For example, consider the user’s statements \(a< b<c<d\). Two possible utilities are: \(u_1(a)=1\), \(u_1(b)=5\), \(u_1(c)=6\), \(u_1(d)=10\), and \(u_2(a)=1\), \(u_2(b)=4\), \(u_2(c)=5\), \(u_2(d)=10\). Then, given \(f=\frac{1}{2}[a]+\frac{1}{2}[d]\) and \(g=\frac{1}{2}[b]+\frac{1}{2}[c]\), it is immediate to see that \(f\cdot u_1< g\cdot u_1\) whereas \(f\cdot u_2> g\cdot u_2\).

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Acknowledgements

This work has been supported by the Italian PRIN project Security Horizons.

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  1. Department of Electrical Engineering and Information Technologies, Università di Napoli “Federico II”, Naples, Italy

    Luigi Sauro

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Sauro, L. On the hierarchical nature of partial preferences over lotteries. Auton Agent Multi-Agent Syst 31, 1467–1505 (2017). https://doi.org/10.1007/s10458-017-9368-6

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