{"intvolume":"        23","citation":{"ista":"Chatterjee K, Majumdar R. 2012. Discounting and averaging in games across time scales. International Journal of Foundations of Computer Science. 23(3), 609–625.","ama":"Chatterjee K, Majumdar R. Discounting and averaging in games across time scales. <i>International Journal of Foundations of Computer Science</i>. 2012;23(3):609-625. doi:<a href=\"https://doi.org/10.1142/S0129054112400308\">10.1142/S0129054112400308</a>","short":"K. Chatterjee, R. Majumdar, International Journal of Foundations of Computer Science 23 (2012) 609–625.","chicago":"Chatterjee, Krishnendu, and Ritankar Majumdar. “Discounting and Averaging in Games across Time Scales.” <i>International Journal of Foundations of Computer Science</i>. World Scientific Publishing, 2012. <a href=\"https://doi.org/10.1142/S0129054112400308\">https://doi.org/10.1142/S0129054112400308</a>.","mla":"Chatterjee, Krishnendu, and Ritankar Majumdar. “Discounting and Averaging in Games across Time Scales.” <i>International Journal of Foundations of Computer Science</i>, vol. 23, no. 3, World Scientific Publishing, 2012, pp. 609–25, doi:<a href=\"https://doi.org/10.1142/S0129054112400308\">10.1142/S0129054112400308</a>.","apa":"Chatterjee, K., &#38; Majumdar, R. (2012). Discounting and averaging in games across time scales. <i>International Journal of Foundations of Computer Science</i>. World Scientific Publishing. <a href=\"https://doi.org/10.1142/S0129054112400308\">https://doi.org/10.1142/S0129054112400308</a>","ieee":"K. Chatterjee and R. Majumdar, “Discounting and averaging in games across time scales,” <i>International Journal of Foundations of Computer Science</i>, vol. 23, no. 3. World Scientific Publishing, pp. 609–625, 2012."},"publist_id":"3326","author":[{"id":"2E5DCA20-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-4561-241X","first_name":"Krishnendu","last_name":"Chatterjee","full_name":"Chatterjee, Krishnendu"},{"full_name":"Majumdar, Ritankar","last_name":"Majumdar","first_name":"Ritankar"}],"doi":"10.1142/S0129054112400308","issue":"3","department":[{"_id":"KrCh"}],"page":"609 - 625","date_updated":"2021-01-12T07:42:35Z","quality_controlled":"1","scopus_import":1,"project":[{"name":"Rigorous Systems Engineering","call_identifier":"FWF","grant_number":"S 11407_N23","_id":"25832EC2-B435-11E9-9278-68D0E5697425"},{"name":"Microsoft Research Faculty Fellowship","_id":"2587B514-B435-11E9-9278-68D0E5697425"}],"_id":"3314","abstract":[{"text":"We introduce two-level discounted and mean-payoff games played by two players on a perfect-information stochastic game graph. The upper level game is a discounted or mean-payoff game and the lower level game is a (undiscounted) reachability game. Two-level games model hierarchical and sequential decision making under uncertainty across different time scales. For both discounted and mean-payoff two-level games, we show the existence of pure memoryless optimal strategies for both players and an ordered field property. We show that if there is only one player (Markov decision processes), then the values can be computed in polynomial time. It follows that whether the value of a player is equal to a given rational constant in two-level discounted or mean-payoff games can be decided in NP ∩ coNP. We also give an alternate strategy improvement algorithm to compute the value. © 2012 World Scientific Publishing Company.","lang":"eng"}],"publication":"International Journal of Foundations of Computer Science","month":"04","day":"01","type":"journal_article","volume":23,"publisher":"World Scientific Publishing","year":"2012","user_id":"3E5EF7F0-F248-11E8-B48F-1D18A9856A87","date_published":"2012-04-01T00:00:00Z","date_created":"2018-12-11T12:02:37Z","oa_version":"None","language":[{"iso":"eng"}],"title":"Discounting and averaging in games across time scales","publication_status":"published","status":"public"}