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Satisficing Versus Optimality: Criteria for Sustainability

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Abstract

Economic analysis addresses risk and long-term issues with discounted expected utility, focusing on optimality. Viability theory is rooted on satisfying sustainability constraints over time, focusing on feasibility. We build a bridge between these two approaches by establishing that viability is equivalent to an array of degenerate intertemporal optimization problems. First, we focus our attention on the deterministic case. We highlight the connections between the viability kernel and the minimum time of crisis. Carrying on, we lay out stochastic viability, turning the spotlight onto the notions of viable scenario and maximal viability probability. Our conceptual results bring the viability approach closer to the economic approach, especially in the stochastic case and regarding efficiency. We discuss the possible use of viability as a theoretical framework for biodiversity conservation, ecosystem management and climate change issues.

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Notes

  1. The word satisficing inspired by Simon (1957) and bounded rationality does not here refer to a situation of limited information in the decision process but as an alternative to optimality.

  2. When the horizon T<+∞, the indicator \(\mathcal{I}^{k}_{T} \) only maps the state space \(\mathbb {X}\) towards \(\mathbb {R}\) because there is no decision at time T. However, for the sake of homogeneity of notations, we will write \(\mathcal{I}^{k}_{T} (x(T),c(T) ) \) for \(\mathcal{I}^{k}_{T} (x(T) ) \).

  3. Without loss of generality, a “bad” indicator, such as pollution, can be represented by its negative value, so that the direction of the inequality holds.

  4. Since the indicator \(\mathcal{I}^{k}_{t}\) and the threshold \(\theta^{k}_{t}\) are allowed to explicitly depend on time t, we can cover the case of absence of constraints (take \(\mathcal{I}^{k}_{t}\) having constant value greater than \(\theta^{k}_{t}\)), or of final target constraint (take \(\mathcal{I}^{k}_{t}(x,c)=0\) and \(\theta^{k}_{t}=0\) for all t=t 0,…,T−1, but not for \(\mathcal{I}^{k}_{T}(x,c)\) and \(\theta^{k}_{T}\)).

  5. Such functions are also called indicator functions. Since we already use the term “indicator” with another meaning, we choose to speak of characteristic functions.

  6. An equivalent additive form is \(\sum_{k=1}^{K} ( 1 - \mathbf{1}_{[\theta^{k}_{t},+\infty[} (\mathcal{I}^{k}_{t} (x(t),c(t) ) ) ) = 0 \), and the minimum form is \(\min_{k=1, \ldots, K} \mathbf{1}_{[\theta^{k}_{t},+\infty[} (\mathcal{I}^{k}_{t} (x(t),c(t) ) ) = 1 \).

  7. As we are summing nonnegative numbers, the given sum is mathematically well-defined whether the time horizon is finite, i.e., T<+∞, or infinite, i.e., T=+∞.

  8. See footnote 7.

  9. As we are multiplying numbers belonging to [0,1], the given product is mathematically well-defined whether the time horizon is finite, i.e., T<+∞, or infinite, i.e., T=+∞.

  10. So does the time-multiplicative form (13).

  11. The probability \(\mathbb {P}\) is defined on the Borel product σ-field of \(\mathbb {W}^{T-t_{0}+1} \). The mappings G t , \(\mathcal{I}^{1}_{t}\), …, \(\mathcal{I}^{k}_{t}\), and all decision rules \(\mathfrak{c}_{t}\) (see below) are supposed to be measurable.

  12. The notation w(⋅)=(w(t 0),…,w(T−1)) still denotes a generic point in Ω; however, it may also be interpreted as a sequence of random variables when w(⋅) is identified with the identity mapping from Ω to Ω.

  13. When the horizon T<+∞, the \(\mathcal{I}^{k}_{T} \) only maps the state space \(\mathbb {X}\times \mathbb {W}\) towards \(\mathbb {R}\) because there is no decision at time T. However, for the sake of homogeneity of notations, we will write \(\mathcal{I}^{k}_{T} (x(T),c(T), w(T) ) \) for \(\mathcal{I}^{k}_{T} (x(T),w(T) ) \).

  14. As a consequence, we do not consider the case where only a corrupted observation of the state is available to the decision-maker (as it may nevertheless be the case in practical situations).

  15. Mathematical materials for stochastic viability can be found in Aubin and Da Prato (1998), Buckdahn et al. (2004) but they focus on the continuous time case. Contributions for discrete time systems are: Doyen et al. (2007), Béné and Doyen (2008), De Lara and Doyen (2008), De Lara and Martinet (2009), Doyen and De Lara (2010).

  16. An exponential increase in relative prices may nevertheless compensate the exponential decrease (Boiteux 1976; Philibert 1999, 2006; Sterner and Persson 2008).

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Acknowledgements

The authors are indebted to the Editor and to two Reviewers for their useful comments.

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Authors and Affiliations

  1. CERMICS (ENPC), Université Paris-Est, 6–8 Avenue Blaise Pascal, Cité Descartes, 77455, Marne-la-Vallée, France

    Michel De Lara

  2. INRA, UMR210 Economie Publique, 78850, Thiverval-Grignon, France

    Vincent Martinet

  3. CNRS, GREThA, University of Bordeaux 4, Avenue Léon Duguit, 33608, Pessac Cedex, France

    Luc Doyen

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  1. Michel De Lara
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Correspondence to Michel De Lara.

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De Lara, M., Martinet, V. & Doyen, L. Satisficing Versus Optimality: Criteria for Sustainability. Bull Math Biol 77, 281–297 (2015). https://doi.org/10.1007/s11538-014-9944-8

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