[{"status":"public","corr_author":"1","author":[{"id":"02d96aae-000e-11ec-b801-cadd0a5eefbb","full_name":"Asadi, Ali","last_name":"Asadi","first_name":"Ali"}],"year":"2026","user_id":"8b945eb4-e2f2-11eb-945a-df72226e66a9","date_updated":"2026-10-12T13:38:13Z","language":[{"iso":"eng"}],"page":"83","degree_awarded":"MS","supervisor":[{"full_name":"Chatterjee, Krishnendu","last_name":"Chatterjee","id":"2E5DCA20-F248-11E8-B48F-1D18A9856A87","first_name":"Krishnendu","orcid":"0000-0002-4561-241X"}],"ec_funded":1,"file_date_updated":"2026-10-12T13:13:51Z","doi_confirm":"1","tmp":{"image":"/images/cc_by.png","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)"},"oa_version":"Published Version","ddc":["000"],"title":"Approximation of values and equilibria in stochastic games: Complexity and algorithms","article_processing_charge":"No","publisher":"Institute of Science and Technology Austria","date_created":"2026-10-08T09:07:53Z","related_material":{"record":[{"id":"17099","relation":"part_of_dissertation","status":"public"},{"id":"21281","relation":"part_of_dissertation","status":"public"}]},"OA_place":"publisher","date_published":"2026-10-09T00:00:00Z","department":[{"_id":"GradSch"},{"_id":"KrCh"}],"has_accepted_license":"1","file":[{"relation":"main_file","date_updated":"2026-10-12T13:13:51Z","file_size":694682,"content_type":"application/pdf","date_created":"2026-10-09T11:12:08Z","creator":"aasadi","access_level":"open_access","file_name":"2026_Asadi_Ali_Thesis.pdf","checksum":"c6b368f342409de884eafdd3af7867c5","file_id":"23081"},{"creator":"aasadi","access_level":"closed","date_created":"2026-10-09T11:12:56Z","file_size":1538723,"content_type":"application/zip","date_updated":"2026-10-09T11:47:47Z","relation":"source_file","file_id":"23082","checksum":"71ecb0b666438f395a92b395e315f904","file_name":"2026_Asadi_Ali_Thesis.zip"}],"doi":"10.15479/AT-ISTA-23079","alternative_title":["ISTA Master’s Thesis"],"citation":{"ama":"Asadi A. Approximation of values and equilibria in stochastic games: Complexity and algorithms. 2026. doi:<a href=\"https://doi.org/10.15479/AT-ISTA-23079\">10.15479/AT-ISTA-23079</a>","ieee":"A. Asadi, “Approximation of values and equilibria in stochastic games: Complexity and algorithms,” Institute of Science and Technology Austria, 2026.","apa":"Asadi, A. (2026). <i>Approximation of values and equilibria in stochastic games: Complexity and algorithms</i>. Institute of Science and Technology Austria. <a href=\"https://doi.org/10.15479/AT-ISTA-23079\">https://doi.org/10.15479/AT-ISTA-23079</a>","ista":"Asadi A. 2026. Approximation of values and equilibria in stochastic games: Complexity and algorithms. Institute of Science and Technology Austria.","short":"A. Asadi, Approximation of Values and Equilibria in Stochastic Games: Complexity and Algorithms, Institute of Science and Technology Austria, 2026.","chicago":"Asadi, Ali. “Approximation of Values and Equilibria in Stochastic Games: Complexity and Algorithms.” Institute of Science and Technology Austria, 2026. <a href=\"https://doi.org/10.15479/AT-ISTA-23079\">https://doi.org/10.15479/AT-ISTA-23079</a>.","mla":"Asadi, Ali. <i>Approximation of Values and Equilibria in Stochastic Games: Complexity and Algorithms</i>. Institute of Science and Technology Austria, 2026, doi:<a href=\"https://doi.org/10.15479/AT-ISTA-23079\">10.15479/AT-ISTA-23079</a>."},"publication_identifier":{"issn":["2791-4585"]},"day":"09","fulldoi":"https://doi.org/10.15479/AT-ISTA-23079","publication_status":"published","type":"dissertation","project":[{"name":"Formal Methods for Stochastic Models: Algorithms and Applications","_id":"0599E47C-7A3F-11EA-A408-12923DDC885E","grant_number":"863818","call_identifier":"H2020"},{"grant_number":"COE12","_id":"4029cfc7-b034-11f1-9e55-88ab2ff3b6ee","name":"Bilateral Artificial Intelligence (Chatterjee)"},{"grant_number":"101020093","call_identifier":"H2020","name":"Vigilant Algorithmic Monitoring of Software","_id":"62781420-2b32-11ec-9570-8d9b63373d4d"}],"_id":"23079","oa":1,"month":"10","acknowledgement":"The research reported in Part I was partially supported by the European Research\r\nCouncil (ERC) Consolidator Grant 863818 (ForM-SMArt), by the Austrian Science Fund\r\n(FWF) 10.55776/COE12, and by the French Agence Nationale de la Recherche (ANR) grant\r\nANR-21-CE40-0020 (CONVERGENCE project). The research reported in Part II is a part of\r\nproject VAMOS, which has received funding from the European Research Council (ERC) under\r\ngrant agreement No.101020093.","abstract":[{"lang":"eng","text":"We study the computational aspects of the two central solution concepts of stochastic games\r\nplayed on finite graphs for an infinite number of steps: the value in two-player zero-sum\r\ngames, and Nash equilibria in multi-player games. In both settings, exact solutions are in\r\ntractable. Fortwo-playerzero-sumconcurrentstochasticgames,valuescanbeirrational,and\r\nthe associated decision problem is SQRT-SUM hard. For multi-player turn-based stochastic\r\ngames,theequilibriumprobabilitiescanbeirrational,anddecidingtheconstrainedexistence\r\nof a Nash equilibrium is undecidable in the presence of memory, and ∃R-complete when the\r\nplayers are restricted to stationary strategies. Hence, in both settings, the natural computational problem is the approximation of the solution within an arbitrary additive error. In\r\nthis thesis, we present improved complexity results and algorithms for such approximation\r\nproblems.\r\nThe first part of the thesis considers two-player zero-sum concurrent stochastic games\r\nwith two classic objectives: (a) stateful-discounted objectives, which are similar to the classic\r\ndiscounted-sum objectives, but states are associated with different discount factors rather\r\nthan a single discount factor; and (b) parity objectives, which are a canonical representation\r\nfor ω-regular objectives. For stateful-discounted objectives, given an ordering of the discount\r\nfactors, the limit value is the limit of the value of the stateful-discounted objectives, as\r\nthe discount factors approach zero according to the given order. The value-approximation\r\nproblem is known to be in EXPSPACE for the limit value of stateful-discounted objectives and\r\nin PSPACEforparityobjectives,andthebest-knownalgorithmsforbothproblemsareatleast\r\nexponential time, with an exponential dependence on the number of states and actions. Our\r\nmain results for the value-approximation problem for the limit value of stateful-discounted\r\nobjectives and parity objectives areas follows: (a)we establish TFNPNP complexity; and (b)we\r\npresent algorithms that improve the dependency on the number of actions in the exponent\r\nfrom linear to logarithmic. In particular, if the number of states is constant, our algorithms\r\nrun in polynomial time.\r\nThe second part of the thesis considers multi-player turn-based stochastic games with\r\nterminal payoffs, where the players are restricted to stationary strategies. We study the\r\napproximate constrained existence problem of Nash equilibria: given a game that has a Nash\r\nequilibrium in which the payoff of every player lies in a given interval, compute a strategy\r\nprofile that is an ε-Nash equilibrium and that satisfies the same constraints up to an additive\r\nerror ε. Our main results are as follows: (a) we present an FNPNP algorithm for the problem;\r\nand(b)we establish that its decision version is NP-hard, already with two players. For the\r\nupperbound,weshowthatifaconstrainedNashequilibriumexists,thenthereexists one\r\nin which the non-zero probabilities are at least an inverse of a double-exponential in the\r\ni\r\ninput size, and that such a strategy profile can be represented, up to an additive error ε, with\r\nfloating-point numbers of polynomial size. We complement the upper bound with a lower\r\nbound on this approach: we present a game with only five players in which every constrained\r\nNash equilibrium requires double-exponentially small probabilities.\r\nThe two parts share a common technical toolkit: bounds on solutions of systems of\r\npolynomial inequalities that arise from real algebraic geometry, the floating-point representation of doubly-exponentially small numbers, and guess-and-verify procedures that resolve\r\nsingle-player games with an NP oracle."}]}]