Abstract
In this paper we study state space search problems where the costs of transitions are uncertain. Cost uncertainty can be due to the existence of several scenarios impacting the entire set of transitions; it can also result from local random factors impacting each transition independently, or from more complex combinations of these two cases. This leads us to consider three different settings for handling cost uncertainty in state space graphs. For each of them, we recall some key properties of first-order and second-order stochastic dominance. Then we propose dominance-based heuristic search algorithms to determine the set of possibly optimal solutions with respect to the expected utility model and Yaari’s model, with and without assuming risk aversion. Finally, to preserve scalability on large-size instances, we adapt these algorithms for the fast determination of an \(\varepsilon \)-covering of the potentially optimal solutions.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
Notes
- 1.
- 2.
For the deep readers, the detailed proofs of our lemmas and theorems can be downloaded at:
- 3.
One can reproduce this paper’s results by using the following C++ programs:
- 4.
In dynamic programming, nodes are referred as states.
- 5.
Assume that the costs of each solution-path is bounded below by 0 and above by \(M\in \mathbb {N}\), and that each integer-valued random variable has the same bounds throughout this paper.
- 6.
That is, \(\prec _{\text {YAA}}\) and \(\prec _{\text {EW}}\) are monotonic with respect to \(\prec _{\text {FSD}}\) and \(\prec _{\text {SSD}}\).
- 7.
Where X and Y are thought as the cost r.v.s of paths of \(\mathcal {P}(s,n)\), and Z of a path of \(\mathcal {P}(n,t)\).
- 8.
One can reproduce this paper’s results by using the following C++ programs:
References
Battuta, I.: The Rihla of Ibn Battuta. Ibn Juzayy (1355)
Bonet, B., Geffner, H.: Faster heuristic search algorithms for planning with uncertainty and full feedback. In: Proceedings of IJCAI, pp. 1233–1238 (2003)
Fu, L., Rilett, L.R.: Expected shortest paths in dynamic and stochastic traffic networks. Transp. Res. 32, 317–322 (1998)
Galand, L., Perny, P.: Search for Choquet-optimal paths under uncertainty. In: Proceedings of UAI, pp. 125–132 (2007)
Gao, Y.: Shortest path problem with uncertain arc lengths. Comput. Math. Appl. 62(6), 2591–2600 (2011)
Hanno: The Voyage of Hanno, commander of the Carthaginians, round the parts of Libya beyond the Pillars of Heracles, which he deposited in the Temple of Kronos. A wall in the temple of Ba’al Hammon in Carathage (500 BC)
Hart, P.E., Nilsson, N.J., Raphael, B.: A formal basis for the heuristic determination of minimum cost paths. IEEE Trans. Syst. Cyb. SSC–4(2), 100–107 (1968)
Loui, R.P.: Optimal paths in graphs with stochastic or multidimensional weights. Commun. ACM 26(9), 670–676 (1983)
Muliere, P., Scarsini, M.: A note on stochastic dominance and inequality measures. J. Econ. Theory 49(2), 314–323 (1989)
Nikolova, E., Brand, M., Karger, D.R.: Optimal route planning under uncertainty. In: ICAPS, vol. 6, pp. 131–141 (2006)
Nilsson, N.J.: Problem-Solving Methods in Artificial Intelligence. McGraw-Hill Pub. Co., New York (1971)
Papadimitriou, C.H., Yannakakis, M.: Shortest paths without a map. In: Ausiello, G., Dezani-Ciancaglini, M., Rocca, S.R.D. (eds.) Automata, Languages and Programming. LNCS, vol. 372, pp. 610–620. Springer, Heidelberg (1989)
Pearl, J.: Heuristics: Intelligent Search Strategies for Computer Problem Solving. Addison-Wesley, Reading (1984)
Perny, P., Spanjaard, O., Storme, L.X.: State space search for risk-averse agents. In: IJCAI, pp. 2353–2358 (2007)
Polo, M.: Devisement du Monde, The Travels of Marco Polo. Rustichello da Pisa (1298)
Puterman, M.L.: Markov Decision Processes: Discrete Stochastic Dynamic Programming, 1st edn. Wiley, New York (1994)
Sauma, R.: History of Mar Yahballaha III and of Rabban Sauma. A clergyman (1281–1317)
Savage, L.J.: The Foundations of Statistics. Wiley, New York (1954)
Von Neumann, J., Morgenstern, O.: Theory of Games and Economic Behavior (60th Anniversary Commemorative Edition). Princeton University Press, Princeton (2007)
Wellman, M., Larson, K., Ford, M., Wurman, P.: Path planning under time-dependent uncertainty. In: Proceedings of the Eleventh Conference on Uncertainty in Artificial Intelligence, pp. 532–539 (1995)
Wurman, P., Wellman, M.: Optimal factory scheduling using stochastic dominance A\(^*\). In: Proceedings of the Twelfth Conference on Uncertainty in Artificial Intelligence, pp. 554–563 (1996)
Yaari, M.E.: The dual theory of choice under risk. Econometrica J. Econometric Soc. 55, 95–115 (1987)
Acknowledgements
I wish to thank Patrice Perny and Olivier Spanjaard for their interesting discussions on the problem, and the reviewers for their comments.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this paper
Cite this paper
Ismaili, A. (2015). State Space Search with Stochastic Costs and Risk Aversion. In: Beierle, C., Dekhtyar, A. (eds) Scalable Uncertainty Management. SUM 2015. Lecture Notes in Computer Science(), vol 9310. Springer, Cham. https://doi.org/10.1007/978-3-319-23540-0_5
Download citation
DOI: https://doi.org/10.1007/978-3-319-23540-0_5
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-23539-4
Online ISBN: 978-3-319-23540-0
eBook Packages: Computer ScienceComputer Science (R0)