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On the Origins of Imperfection and Apparent Non-rationality

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Decision Making: Uncertainty, Imperfection, Deliberation and Scalability

Part of the book series: Studies in Computational Intelligence ((SCI,volume 538))

Abstract

Decision making (DM) is a preferences-driven choice among available actions. Under uncertainty, Savage’s axiomatisation singles out Bayesian DM as the adequate normative framework. It constructs strategies generating the optimal actions, while assuming that the decision maker rationally tries to meet her preferences. Descriptive DM theories have observed numerous deviations of the real DM from normative recommendations. The explanation of decision-makers’ imperfection or non-rationality, possibly followed by rectification, is the focal point of contemporary DM research. This chapter falls into this stream and claims that the neglecting a part of the behaviour of the closed DM loop is the major cause of these deviations. It inspects DM subtasks in which this claim matters and where its consideration may practically help. It deals with: (i) the preference elicitation; (ii) the “non-rationality” caused by the difference of preferences declared and preferences followed; (iii) the choice of proximity measures in knowledge and preferences fusion; (iv) ways to a systematic design of approximate DM; and (v) the control of the deliberation effort spent on a DM task via sequential DM. The extent of the above list indicates that the discussion offers more open questions than answers, however, their consideration is the key element of this chapter. Their presentation is an important chapter’s ingredient.

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Notes

  1. 1.

    Savage [68] calls it a small world. Alternative terms like system, plant, object are used.

  2. 2.

    A pd is the Radon-Nikodým derivative of a probabilistic, randomness-modelling measure.

  3. 3.

    The KLD has many names. Relative entropy and cross entropy [70] are the most common.

  4. 4.

    When ignorance includes non-constant internals, Bayesian learning used below becomes stochastic filtering [30]. If moreover, the decision maker’s preferences depend on an action-dependent internal state, the stochastic control problem arises [41]. This general case is not treated here as it complicates explanations without offering any conceptual shift.

  5. 5.

    Further on, the superscript \(^{\star }\) marks pds and actions arising from this ideal closed-loop model.

  6. 6.

    The quest for simple final formulas has motivated a slightly non-standard choice of the “directions” of the ordering operators \(\preceq \), \(\ge \) and \(\prec \), \(>\).

  7. 7.

    The existence of such pairs can be assumed without loss of generality. Indeed, no non-trivial decision task arises if all comparable pairs of behaviours in the original decision-maker-specified partial ordering are equivalent.

  8. 8.

    The mapping \({\mathsf {R}}_{{\mathsf {S}}}\) is common to decision makers differing only in preferences among behaviours.

  9. 9.

    The functional is local if its value on \({\mathsf {\Lambda }}\), artificially written as the sum \({\mathsf {\Lambda }}_{1}+{\mathsf {\Lambda }}_{2}\) of functions \({\mathsf {\Lambda }}_{1},\,{\mathsf {\Lambda }}_{2}\) fulfilling \({\mathsf {\Lambda }}_{1}{\mathsf {\Lambda }}_{2}=0\), is the sum of its values on \({\mathsf {\Lambda }}_{1}\) and \({\mathsf {\Lambda }}_{2}\).

  10. 10.

    The measure serves to all DM tasks facing the same uncertainty. The function \({\mathsf {U}}\) models risk awareness, neutrality or proneness. The function \({\mathsf {U}}\), \({\mathsf {C}}\)-almost surely increasing in its first argument, guarantees that the optimal strategy \({\mathsf {S}}^{o}\) selected from the considered subset of \({\pmb {{{{\mathsf {S}}}}}}\) is not dominated It means that it cannot happen that within this subset there is a strategy \({\mathsf {S}}^{d}\) such that \({\mathsf {\Lambda }}_{{\mathsf {S}}^{d}}(u)\le {\mathsf {\Lambda }}_{{\mathsf {S}}^{o}}(u)\) on \({\pmb {{{u}}}}\) with the sharp inequality on a subset of \({\pmb {{{u}}}}\) of a positive \({\mathsf {C}}\) measure.

  11. 11.

    Giarlotta and Greco [22] represents non-Bayesian set-ups dealing with sets of orderings without a quest for a unique completion.

  12. 12.

    A decision maker interacts with customers in order to influence them in a desirable direction, for instance, to buy a specific product or services. However, even the form of the questionnaire influences the customers: typically, two different ways of posing logically the same question often provide quite different answers. This quantum-mechanics-like effect should be properly modelled.

  13. 13.

    The vast majority of complex technological processes, which should be modelled by high-dimensional nonlinear stochastic partial differential equations with non-smooth boundary conditions, are controlled by proportional-integral-derivative controllers corresponding to simple linear, second order difference equations used as the environment model.

  14. 14.

    The adopted notation \(a^{\star }\) stresses that this action value serves for the construction of \({\mathsf {C}}^{\star }\).

  15. 15.

    \({\mathsf {S}}^{\star }_{0}(a_{t}|o_{t},k_{t-1})\) and \(a^{\star }_{t}(k_{t-1})\) are independent of \(o_{t}\), i.e. \({\mathsf {S}}^{\star }_{}(a_{t}|o_{t},k_{t-1})={\mathsf {S}}^{\star }(a_{t}|k_{t-1})\), see (3.21).

  16. 16.

    The condition \(z_{t}=1\) stresses that the optimisation is performed: it is not stopped.

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Acknowledgments

The reported research has been supported by GAČR 13-13502S. Dr. Anthony Quinn from Trinity College, Dublin, has provided us useful feedback.

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  1. Institute of Information Theory and Automation, The Czech Academy of Sciences, Pod Vodárenskou věží 4, 182 08, Prague 8, Czech Republic

    Miroslav Kárný & Tatiana V. Guy

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  1. Miroslav Kárný
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Correspondence to Miroslav Kárný .

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  1. Institute of Information Theory and Automation, The Czech Academy of Sciences, Prague, Czech Republic

    Tatiana V. Guy

  2. Institute of Information Theory and Automation, The Czech Academy of Sciences, Prague, Czech Republic

    Miroslav Kárný

  3. Santa Fe Institute, Santa Fe, New Mexico, USA

    David H. Wolpert

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Kárný, M., Guy, T.V. (2015). On the Origins of Imperfection and Apparent Non-rationality. In: Guy, T., Kárný, M., Wolpert, D. (eds) Decision Making: Uncertainty, Imperfection, Deliberation and Scalability. Studies in Computational Intelligence, vol 538. Springer, Cham. https://doi.org/10.1007/978-3-319-15144-1_3

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