{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,17]],"date-time":"2026-01-17T04:27:46Z","timestamp":1768624066527,"version":"3.49.0"},"reference-count":22,"publisher":"Springer Science and Business Media LLC","issue":"4","license":[{"start":{"date-parts":[[2021,3,2]],"date-time":"2021-03-02T00:00:00Z","timestamp":1614643200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2021,3,2]],"date-time":"2021-03-02T00:00:00Z","timestamp":1614643200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Dyn Games Appl"],"published-print":{"date-parts":[[2021,12]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>We study subgame <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\phi $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>\u03d5<\/mml:mi>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>-maxmin strategies in two-player zero-sum stochastic games with a countable state space, finite action spaces, and a bounded and universally measurable payoff function. Here, <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\phi $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>\u03d5<\/mml:mi>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> denotes the tolerance function that assigns a nonnegative tolerated error level to every subgame. Subgame <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\phi $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>\u03d5<\/mml:mi>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>-maxmin strategies are strategies of the maximizing player that guarantee the lower value in every subgame within the subgame-dependent tolerance level as given by <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\phi $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>\u03d5<\/mml:mi>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. First, we provide necessary and sufficient conditions for a strategy to be a subgame <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\phi $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>\u03d5<\/mml:mi>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>-maxmin strategy. As a special case, we obtain a characterization for subgame maxmin strategies, i.e., strategies that exactly guarantee the lower value at every subgame. Secondly, we present sufficient conditions for the existence of a subgame <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\phi $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>\u03d5<\/mml:mi>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>-maxmin strategy. Finally, we show the possibly surprising result that each game admits a strictly positive tolerance function <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\phi ^*$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msup>\n                    <mml:mi>\u03d5<\/mml:mi>\n                    <mml:mo>\u2217<\/mml:mo>\n                  <\/mml:msup>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> with the following property: if a player has a subgame <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\phi ^*$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msup>\n                    <mml:mi>\u03d5<\/mml:mi>\n                    <mml:mo>\u2217<\/mml:mo>\n                  <\/mml:msup>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>-maxmin strategy, then he has a subgame maxmin strategy too. As a consequence, the existence of a subgame <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\phi $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>\u03d5<\/mml:mi>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>-maxmin strategy for every positive tolerance function <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\phi $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>\u03d5<\/mml:mi>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> is equivalent to the existence of a subgame maxmin strategy.<\/jats:p>","DOI":"10.1007\/s13235-021-00378-z","type":"journal-article","created":{"date-parts":[[2021,3,2]],"date-time":"2021-03-02T03:02:57Z","timestamp":1614654177000},"page":"704-737","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":7,"title":["Subgame Maxmin Strategies in Zero-Sum Stochastic Games with Tolerance Levels"],"prefix":"10.1007","volume":"11","author":[{"given":"J\u00e1nos","family":"Flesch","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1100-8601","authenticated-orcid":false,"given":"P. Jean-Jacques","family":"Herings","sequence":"additional","affiliation":[]},{"given":"Jasmine","family":"Maes","sequence":"additional","affiliation":[]},{"given":"Arkadi","family":"Predtetchinski","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2021,3,2]]},"reference":[{"key":"378_CR1","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/j.spl.2014.01.009","volume":"88","author":"A Abate","year":"2014","unstructured":"Abate A, Redig F, Tkachev I (2014) On the effect of perturbation of conditional probabilities in total variation. Stat Probab Lett 88:1\u20138","journal-title":"Stat Probab Lett"},{"key":"378_CR2","doi-asserted-by":"publisher","first-page":"159","DOI":"10.1214\/aoms\/1177698513","volume":"39","author":"D Blackwell","year":"1968","unstructured":"Blackwell D, Ferguson T (1968) The big match. Ann Math Stat 39:159\u2013163","journal-title":"Ann Math Stat"},{"key":"378_CR3","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-540-34514-5","volume-title":"Measure Theory","author":"VI Bogachev","year":"2007","unstructured":"Bogachev VI (2007) Measure Theory, vol II. Springer, Berlin"},{"key":"378_CR4","doi-asserted-by":"crossref","unstructured":"Bruy\u00e8re V (2017) Computer aided synthesis: a game-theoretic approach. In: Charlier E, Leroy J, Rigo M (eds) Developments in Language Theory, vol 10396. Lecture Notes in Computer Science. Springer, pp 3\u201335","DOI":"10.1007\/978-3-319-62809-7_1"},{"key":"378_CR5","doi-asserted-by":"crossref","unstructured":"Flesch J, Herings PJJ, Maes J, Predtetchinski A (2018) Subgame maxmin strategies in zero-sum stochastic games with tolerance levels. GSBE Research Memorandum 18\/20, Maastricht University, Maastricht, pp. 1-37","DOI":"10.2139\/ssrn.3237537"},{"key":"378_CR6","doi-asserted-by":"publisher","first-page":"523","DOI":"10.1007\/s00182-015-0468-8","volume":"45","author":"J Flesch","year":"2016","unstructured":"Flesch J, Predtetchinski A (2016) On refinements of subgame perfect $$\\epsilon $$-equilibrium. Int J Game Theory 45:523\u2013542","journal-title":"Int J Game Theory"},{"key":"378_CR7","doi-asserted-by":"publisher","first-page":"728","DOI":"10.1017\/jpr.2018.47","volume":"55","author":"J Flesch","year":"2018","unstructured":"Flesch J, Predtetchinski A, Sudderth W (2018) Characterization and simplification of optimal strategies in positive stochastic games. J Appl Probab 55:728\u2013741","journal-title":"J Appl Probab"},{"key":"378_CR8","doi-asserted-by":"crossref","unstructured":"Flesch J, Thuijsman F, Vrieze OJ (1998) Improving strategies in stochastic games. In: Proceedings of the 37th IEEE Conference on Decision and Control, Vol 3, IEEE, pp. 2674-2679","DOI":"10.1109\/CDC.1998.757857"},{"key":"378_CR9","doi-asserted-by":"crossref","unstructured":"Gillette D (1957) Stochastic games with zero stop probabilities. In: Dresher M, Tucker AW, Wolfe P (eds) Contributions to the Theory of Games, vol III. Annals of Mathematics Studies, Volume 39. Princeton University Press, Princeton, New Jersey, pp 179\u2013187","DOI":"10.1515\/9781400882151-011"},{"key":"378_CR10","doi-asserted-by":"publisher","first-page":"162","DOI":"10.1007\/s13235-012-0054-7","volume":"3","author":"R Laraki","year":"2013","unstructured":"Laraki R, Maitra A, Sudderth W (2013) Two-person zero-sum stochastic games with semicontinuous payoff. Dyn Games Appl 3:162\u2013171","journal-title":"Dyn Games Appl"},{"key":"378_CR11","doi-asserted-by":"publisher","first-page":"126","DOI":"10.1016\/j.geb.2005.05.002","volume":"53","author":"G Mailath","year":"2005","unstructured":"Mailath G, Postlewaite A, Samuelson L (2005) Contemporaneous perfect epsilon-equilibria. Games Econ Behav 53:126\u2013140","journal-title":"Games Econ Behav"},{"key":"378_CR12","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-4002-0","volume-title":"Discrete gambling and stochastic games","author":"A Maitra","year":"1996","unstructured":"Maitra A, Sudderth W (1996) Discrete gambling and stochastic games. Springer, New York"},{"key":"378_CR13","doi-asserted-by":"publisher","first-page":"257","DOI":"10.1007\/s001820050071","volume":"27","author":"A Maitra","year":"1998","unstructured":"Maitra A, Sudderth W (1998) Finitely additive stochastic games with Borel measurable payoffs. Int J Game Theory 27:257\u2013267","journal-title":"Int J Game Theory"},{"key":"378_CR14","doi-asserted-by":"publisher","first-page":"1565","DOI":"10.2307\/2586667","volume":"63","author":"DA Martin","year":"1998","unstructured":"Martin DA (1998) The determinacy of Blackwell games. J Symb Logic 63:1565\u20131581","journal-title":"J Symb Logic"},{"key":"378_CR15","doi-asserted-by":"publisher","first-page":"120","DOI":"10.1007\/s13235-014-0122-2","volume":"5","author":"A Mashiah-Yaakovi","year":"2015","unstructured":"Mashiah-Yaakovi A (2015) Correlated equilibria in stochastic games with Borel measurable payoffs. Dyn Games Appl 5:120\u2013135","journal-title":"Dyn Games Appl"},{"key":"378_CR16","doi-asserted-by":"publisher","DOI":"10.1002\/9780470316887","volume-title":"Markov decision processes, discrete stochastic dynamic programming","author":"ML Puterman","year":"1994","unstructured":"Puterman ML (1994) Markov decision processes, discrete stochastic dynamic programming. Wiley, Hoboken"},{"key":"378_CR17","doi-asserted-by":"publisher","first-page":"136","DOI":"10.1016\/0022-0531(80)90037-X","volume":"22","author":"R Radner","year":"1980","unstructured":"Radner R (1980) Collusive behavior in noncooperative epsilon-equilibria of oligopolies with long but finite lives. J Econ Theory 22:136\u2013154","journal-title":"J Econ Theory"},{"key":"378_CR18","doi-asserted-by":"publisher","first-page":"433","DOI":"10.1007\/PL00008766","volume":"119","author":"D Rosenberg","year":"2001","unstructured":"Rosenberg D, Solan E, Vieille N (2001) Stopping games with randomized strategies. Probab Theory Related Fields 119:433\u2013451","journal-title":"Probab Theory Related Fields"},{"key":"378_CR19","first-page":"301","volume":"121","author":"R Selten","year":"1965","unstructured":"Selten R (1965) Spieltheoretische Behandlung eines Oligopolmodells mit Nachfragetr\u00e4gheit: Teil I: Bestimmung des dynamischen Preisgleichgewichts. Zeitschrift f\u00fcr die gesamte Staatswissenschaft 121:301\u2013324","journal-title":"Zeitschrift f\u00fcr die gesamte Staatswissenschaft"},{"key":"378_CR20","first-page":"1185","volume":"12","author":"E Solan","year":"2002","unstructured":"Solan E, Vieille N (2002) Uniform value in recursive games. Ann App Probab 12:1185\u20131201","journal-title":"Ann App Probab"},{"key":"378_CR21","doi-asserted-by":"publisher","first-page":"453","DOI":"10.1126\/science.7455683","volume":"211","author":"A Tversky","year":"1981","unstructured":"Tversky A, Kahneman D (1981) The framing of decisions and the psychology of choice. Science 211:453\u2013458","journal-title":"Science"},{"key":"378_CR22","doi-asserted-by":"publisher","DOI":"10.1142\/2948","volume-title":"Martingales and stochastic analysis","author":"J Yeh","year":"1995","unstructured":"Yeh J (1995) Martingales and stochastic analysis, vol 1. World Scientific, Singapore"}],"container-title":["Dynamic Games and Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s13235-021-00378-z.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s13235-021-00378-z\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s13235-021-00378-z.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,10,24]],"date-time":"2021-10-24T13:30:07Z","timestamp":1635082207000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s13235-021-00378-z"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,3,2]]},"references-count":22,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2021,12]]}},"alternative-id":["378"],"URL":"https:\/\/doi.org\/10.1007\/s13235-021-00378-z","relation":{},"ISSN":["2153-0785","2153-0793"],"issn-type":[{"value":"2153-0785","type":"print"},{"value":"2153-0793","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,3,2]]},"assertion":[{"value":"4 February 2021","order":1,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"2 March 2021","order":2,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}