Sethi-Skiba point

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Sethi-Skiba points,[1][2][3][4] also known as DNSS points, arise in optimal control problems that exhibit multiple optimal solutions. A Sethi-Skiba point is an indifference point in an optimal control problem such that starting from such a point, the problem has more than one different optimal solutions. A good discussion of such points can be found in Grass et al.[2][5][6]

Definition

Of particular interest here are discounted infinite horizon optimal control problems that are autonomous.[2] These problems can be formulated as

[math]\displaystyle{ \max_{u(t)\in \Omega}\int_0^{\infty} e^{-\rho t} \varphi\left(x(t), u(t)\right)dt }[/math]

s.t.

[math]\displaystyle{ \dot{x}(t) = f\left(x(t), u(t)\right), x(0) = x_{0}, }[/math]

where [math]\displaystyle{ \rho \gt 0 }[/math] is the discount rate, [math]\displaystyle{ x(t) }[/math] and [math]\displaystyle{ u(t) }[/math] are the state and control variables, respectively, at time [math]\displaystyle{ t }[/math], functions [math]\displaystyle{ \varphi }[/math] and [math]\displaystyle{ f }[/math] are assumed to be continuously differentiable with respect to their arguments and they do not depend explicitly on time [math]\displaystyle{ t }[/math], and [math]\displaystyle{ \Omega }[/math] is the set of feasible controls and it also is explicitly independent of time [math]\displaystyle{ t }[/math]. Furthermore, it is assumed that the integral converges for any admissible solution [math]\displaystyle{ \left(x(.), u(.)\right) }[/math]. In such a problem with one-dimensional state variable [math]\displaystyle{ x }[/math], the initial state [math]\displaystyle{ x_{0} }[/math] is called a Sethi-Skiba point if the system starting from it exhibits multiple optimal solutions or equilibria. Thus, at least in the neighborhood of [math]\displaystyle{ x_0 }[/math], the system moves to one equilibrium for [math]\displaystyle{ x \gt x_0 }[/math] and to another for [math]\displaystyle{ x \lt x_0 }[/math]. In this sense, [math]\displaystyle{ x_0 }[/math] is an indifference point from which the system could move to either of the two equilibria.

For two-dimensional optimal control problems, Grass et al.[5] and Zeiler et al.[7] present examples that exhibit DNSS curves.

Some references on the applications of Sethi-Skiba points are Caulkins et al.[8], Zeiler et al.[9], and Carboni and Russu[10]

History

Suresh P. Sethi identified such indifference points for the first time in 1977.[11] Further, Skiba,[12] Sethi,[13][14][15] and Deckert and Nishimura[16] explored these indifference points in economic models. The term DNSS (Deckert, Nishimura, Sethi, Skiba) points, introduced by Grass et al.,[5] recognizes (alphabetically) the contributions of these authors. These indifference points have been also referred to as Skiba points or DNS points in earlier literature.[5]

Example

A simple problem exhibiting this behavior is given by [math]\displaystyle{ \varphi\left(x,u\right) =xu, }[/math] [math]\displaystyle{ f\left(x,u\right) = -x + u, }[/math] and [math]\displaystyle{ \Omega = \left[-1, 1\right] }[/math]. It is shown in Grass et al.[5] that [math]\displaystyle{ x_{0} = 0 }[/math] is a Sethi-Skiba point for this problem because the optimal path [math]\displaystyle{ x(t) }[/math] can be either [math]\displaystyle{ \left(1-e^{-t}\right) }[/math] or [math]\displaystyle{ \left(-1+e^{-t}\right) }[/math]. Note that for [math]\displaystyle{ x_{0} \lt 0 }[/math], the optimal path is [math]\displaystyle{ x(t) = -1 + e^{-t\left(x_{0}+1 \right)} }[/math] and for [math]\displaystyle{ x_{0} \gt 0 }[/math], the optimal path is [math]\displaystyle{ x(t) = 1 + e^{-t\left(x_{0}-1 \right)} }[/math].

Extensions

For further details and extensions, the reader is referred to Grass et al.[5]

References

  1. ↑ Caulkins, Jonathan P.; Grass, Dieter; Feichtinger, Gustav; Hartl, Richard F.; Kort, Peter M.; Prskawetz, Alexia; Seidl, Andrea; Wrzaczek, Stefan (2021-03-01). "The optimal lockdown intensity for COVID-19" (in en). Journal of Mathematical Economics. The economics of epidemics and emerging diseases 93: 102489. doi:10.1016/j.jmateco.2021.102489. ISSN 0304-4068. https://www.sciencedirect.com/science/article/pii/S0304406821000276. 
  2. ↑ 2.0 2.1 2.2 Sethi, Suresh P. (2021). "Optimal Control Theory" (in en-gb). Sethi, S.P. (2021). " Optimal Control Theory: Applications to Management Science and Economics". Fourth Edition, Springer Nature Switzerland AG,ISBN 978-3-319-98236-6. doi:10.1007/978-3-319-98237-3. ISBN 978-3-319-98236-6. https://doi.org/10.1007/978-3-319-98237-3. 
  3. ↑ Caulkins, Jonathan P.; Grass, Dieter; Feichtinger, Gustav; Hartl, Richard F.; Kort, Peter M.; Prskawetz, Alexia; Seidl, Andrea; Wrzaczek, Stefan (2022), Boado-Penas, María del Carmen; Eisenberg, Julia; Şahin, Şule, eds., "COVID-19 and Optimal LockdownStrategies: The Effect of New and MoreVirulent Strains" (in en), Pandemics: Insurance and Social Protection, Springer Actuarial (Cham: Springer International Publishing): pp. 163–190, doi:10.1007/978-3-030-78334-1_9, ISBN 978-3-030-78334-1 
  4. ↑ Caulkins, Jonathan; Grass, Dieter; Feichtinger, Gustav; Hartl, Richard; Kort, Peter M.; Prskawetz, Alexia; Seidl, Andrea; Wrzaczek, Stefan (2020-12-02). "How long should the COVID-19 lockdown continue?" (in en). PLOS ONE 15 (12): e0243413. doi:10.1371/journal.pone.0243413. ISSN 1932-6203. PMID 33264368. 
  5. ↑ 5.0 5.1 5.2 5.3 5.4 5.5 Grass, D.; Caulkins, J. P.; Feichtinger, G.; Tragler, G.; Behrens, D. A. (2008). Optimal Control of Nonlinear Processes: With Applications in Drugs, Corruption, and Terror. Springer. ISBN 978-3-540-77646-8. 
  6. ↑ Caulkins, J. P., Grass, D., Feichtinger, G., Hartl, R. F., Kort, P. M., Prskawetz, A., Seidl, A., Wrzaczek, A. (2020). “When should the Covid-19 lockdown end?”. OR News, Ausgabe 69: 10-13
  7. ↑ Zeiler, I., Caulkins, J., Grass, D., Tragler, G. (2009). Keeping Options Open: An Optimal Control Model with Trajectories that Reach a DNSS Point in Positive Time. SIAM Journal on Control and Optimization, Vol. 48, No. 6, pp. 3698-3707.| doi =10.1137/080719741 |
  8. ↑ Caulkins, J. P.; Feichtinger, G.; Grass, D.; Tragler, G. (2009). "Optimal control of terrorism and global reputation: A case study with novel threshold behavior". Operations Research Letters 37 (6): 387–391. doi:10.1016/j.orl.2009.07.003. 
  9. ↑ I. Zeiler, J. P. Caulkins, and G. Tragler. When Two Become One: Optimal Control of Interacting Drug. Working paper, Vienna University of Technology, Vienna, Austria
  10. ↑ Carboni, Oliviero A.; Russu, Paolo (2021-06-01). "Taxation, Corruption and Punishment: Integrating Evolutionary Game into the Optimal Control of Government Policy". International Game Theory Review 23 (2): 2050019. doi:10.1142/S021919892050019X. ISSN 0219-1989. https://www.worldscientific.com/doi/abs/10.1142/S021919892050019X. 
  11. ↑ Sethi, S.P. (1977). "Nearest Feasible Paths in Optimal Control Problems: Theory, Examples, and Counterexamples". Journal of Optimization Theory and Applications 23 (4): 563–579. doi:10.1007/BF00933297. 
  12. ↑ Skiba, A.K. (1978). "Optimal Growth with a Convex-Concave Production Function". Econometrica 46 (3): 527–539. doi:10.2307/1914229. 
  13. ↑ Sethi, S. P. (1977-12-01). "Nearest feasible paths in optimal control problems: Theory, examples, and counterexamples" (in en). Journal of Optimization Theory and Applications 23 (4): 563–579. doi:10.1007/BF00933297. ISSN 1573-2878. https://doi.org/10.1007/BF00933297. 
  14. ↑ Sethi, S.P. (1979). "Optimal Advertising Policy with the Contagion Model". Journal of Optimization Theory and Applications 29 (4): 615–627. doi:10.1007/BF00934454. 
  15. ↑ Sethi, S.P., "Optimal Quarantine Programmes for Controlling an Epidemic Spread," Journal of Operational Research Society, 29(3), 1978, 265-268. JSTOR 3009454 SSRN 3587573
  16. ↑ Deckert, D.W.; Nishimura, K. (1983). "A Complete Characterization of Optimal Growth Paths in an Aggregated Model with Nonconcave Production Function". Journal of Economic Theory 31 (2): 332–354. doi:10.1016/0022-0531(83)90081-9. 



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