Sethi-Skiba point

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

${\displaystyle \max _{u(t)\in \Omega }\int _{0}^{\infty }e^{-\rho t}\varphi \left(x(t),u(t)\right)dt}$

s.t.

${\displaystyle {\dot {x}}(t)=f\left(x(t),u(t)\right),x(0)=x_{0},}$

where ${\displaystyle \rho >0}$ is the discount rate, ${\displaystyle x(t)}$ and ${\displaystyle u(t)}$ are the state and control variables, respectively, at time ${\displaystyle t}$, functions ${\displaystyle \varphi }$ and ${\displaystyle f}$ are assumed to be continuously differentiable with respect to their arguments and they do not depend explicitly on time ${\displaystyle t}$, and ${\displaystyle \Omega }$ is the set of feasible controls and it also is explicitly independent of time ${\displaystyle t}$. Furthermore, it is assumed that the integral converges for any admissible solution ${\displaystyle \left(x(.),u(.)\right)}$. In such a problem with one-dimensional state variable ${\displaystyle x}$, the initial state ${\displaystyle x_{0}}$ 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 ${\displaystyle x_{0}}$, the system moves to one equilibrium for ${\displaystyle x>x_{0}}$ and to another for ${\displaystyle x<x_{0}}$. In this sense, ${\displaystyle x_{0}}$ 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 ${\displaystyle \varphi \left(x,u\right)=xu,}$${\displaystyle f\left(x,u\right)=-x+u,}$ and ${\displaystyle \Omega =\left[-1,1\right]}$. It is shown in Grass et al. ^[5] that ${\displaystyle x_{0}=0}$ is a Sethi-Skiba point for this problem because the optimal path ${\displaystyle x(t)}$ can be either ${\displaystyle \left(1-e^{-t}\right)}$ or ${\displaystyle \left(-1+e^{-t}\right)}$. Note that for ${\displaystyle x_{0}<0}$, the optimal path is ${\displaystyle x(t)=-1+e^{-t\left(x_{0}+1\right)}}$ and for ${\displaystyle x_{0}>0}$, the optimal path is ${\displaystyle x(t)=1+e^{-t\left(x_{0}-1\right)}}$.

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". 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.
2. 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. doi: 10.1371/journal.pone.0243413 . ISSN   1932-6203. PMC   7710360 . PMID   33264368.
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.
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.