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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

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

s.t.

where is the discount rate, and are the state and control variables, respectively, at time , functions and are assumed to be continuously differentiable with respect to their arguments and they do not depend explicitly on time , and is the set of feasible controls and it also is explicitly independent of time . Furthermore, it is assumed that the integral converges for any admissible solution . In such a problem with one-dimensional state variable , the initial state 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 , the system moves to one equilibrium for and to another for . In this sense, 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

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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

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A simple problem exhibiting this behavior is given by and . It is shown in Grass et al.[5] that is a Sethi-Skiba point for this problem because the optimal path can be either or . Note that for , the optimal path is and for , the optimal path is .

Extensions

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

References

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  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". Journal of Mathematical Economics. The economics of epidemics and emerging diseases. 93: 102489. doi:10.1016/j.jmateco.2021.102489. hdl:10067/1777560151162165141. ISSN 0304-4068. PMC 7857053. PMID 33558783.
  2. ^ a b c Sethi, Suresh P. (2021). "Optimal Control Theory". 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.
  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", Pandemics: Insurance and Social Protection, Springer Actuarial, Cham: Springer International Publishing, pp. 163–190, doi:10.1007/978-3-030-78334-1_9, hdl:10419/229887, 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?". PLOS ONE. 15 (12): e0243413. Bibcode:2020PLoSO..1543413C. doi:10.1371/journal.pone.0243413. ISSN 1932-6203. PMC 7710360. PMID 33264368.
  5. ^ a b c d e f 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.
  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. S2CID 123705828.
  12. ^ Skiba, A.K. (1978). "Optimal Growth with a Convex-Concave Production Function". Econometrica. 46 (3): 527–539. doi:10.2307/1914229. JSTOR 1914229.
  13. ^ Sethi, S. P. (1977-12-01). "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. ISSN 1573-2878.
  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. S2CID 121398518.
  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.