康尼島（英語：），又译科尼島，是位於美國紐約市布魯克林區的半島，原本為一座海島，其面向大西洋的海灘是美國知名的休閒娛樂區域。居民大多集中位於半島的西側，約有六萬人左右，範圍西至希捷社區，東至布萊登海灘和曼哈頓海灘，而北至葛瑞福山德社區。 二十世紀前葉在美國極為知名的太空星際樂園即是以康尼島作為主要的腹地，該樂園在二次大戰後開始衰退，並持續荒廢了許久。在最近幾年，康尼島因為凱斯班公園的開幕而重新繁榮起來，凱斯班公園是職棒小聯盟球隊布魯克林旋風的主要球場。旋風隊在當地十分受到歡迎，每季開賽時都會吸引許多球迷到場觀戰。 ..

Anjos da guarda são os anjos que segundo as crenças cristãs, Deus envia no nosso nascimento para nos proteger durante toda a nossa vida. Argumenta-se que a Bíblia sustenta em algumas ocasiões a crença do anjo da guarda: "Vou enviar um anjo adiante de ti para ..

Altay Cumhuriyeti (Rusça: Респу́блика Алта́й / Respublika Altay; Altay Türkçesi: Алтай Республика / Altay Respublika), Rusya'nın en güneyinde yer alan, federasyona bağlı bir özerk cumhuriyet. Orta Asya'da Asya kıtasının coğrafî merkezinin hemen kuzeyinde ve ..

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希西家王 (希伯來語：，英語：）是猶大末年的君主，也是猶大國歷史中極尊重上帝的君王，在位29年。終年54歲。他在位的年份有兩種說法：其一是前715年-前687年；另一種是前716年-前687年。他的德行在其前後的猶大列王中，没有一個能及他。其希伯來名字的意思是“被神加力量”。 希西家的父親亞哈斯是一個背逆上帝的君王。因此在希西家當政之初的猶大國，無論政治，宗教上都极其黑暗。根據《聖經》記載，因为北國以色列被亞述攻滅，亞述王可以趁勢来攻打猶大國；又猶大的先王亞哈斯曾封鎖了聖殿之路，引導舉國崇拜偶像，大大得罪上帝。若非上帝的憐憫，為了堅定向大衛家所說的應許，猶大國的暫得幸存。希西家在二十五歲就登基作王，且正在國家危急之秋，由於行耶和華上帝眼中看為正的事，因而得上帝的憐憫，得以成功脫離亞述大軍的攻擊和一場致死的大病。他樂於聽從當代先知以賽亞的指導，使他為上帝大發熱心。 ..

The OnePlus 2 (also abbreviated as OP2) is a smartphone designed by OnePlus. It is the successor to the OnePlus One. OnePlus revealed the phone on 28 July 2015 via virtual reality, using Google's Cardboard visor and their own app. OnePlus sold out 30,000 units ..

兴隆街镇，是中华人民共和国四川省内江市资中县下辖的一个乡镇级行政单位。 兴隆街镇下辖以下地区： 兴隆街社区、兴松村、玄天观村、三元村、金星村、三皇庙村、双桥村、红庙子村、华光村、高峰村、芦茅湾村、篮家坝村、五马村和解放村。

Национальная и университетская библиотека (словен. Narodna in univerzitetna knjižnica, NUK), основанная в 1774 году, — один из важнейших образовательных и культурных учреждений Словении. Она располагается в центре столицы Любляна, между улицами Турьяшка (Turjaška ..

Mauser M1924 (или M24) — серия винтовок компании Mauser, использовавшихся в армиях Бельгии и Югославии. Внешне напоминают чехословацкие винтовки vz. 24, в которых использовались стандартный открытый прицел, патроны калибра 7,92×57 мм (или 8×57 мм), укороченные ..

第三条道路（英語：），又称新中间路线（Middle Way），是一种走在自由放任资本主义和传统社会主义中间的一种政治经济理念的概称。它由中间派所倡导，是社会民主主义的一个流派，英国工党称其为「现代化的社会民主主义」。它的中心思想是既不主张纯粹的自由市场，也不主张纯粹的社會主義，主张在两者之间取折衷方案。 第三条道路不只单单是走在中间，或只是一种妥协或混合出来的东西，第三条道路的提倡者看到了社会主义和资本主义互有不足之处，所以偏向某一极端也不是一件好事，第三条道路正正是揉合了双方主义的优点，互补不足而成的政治哲学。 ..

In game theory, a **stochastic game**, introduced by Lloyd Shapley in the early 1950s, is a dynamic game with **probabilistic transitions** played by one or more players. The game is played in a sequence of stages. At the beginning of each stage the game is in some **state**. The players select actions and each player receives a **payoff** that depends on the current state and the chosen actions. The game then moves to a new random state whose distribution depends on the previous state and the actions chosen by the players. The procedure is repeated at the new state and play continues for a finite or infinite number of stages. The total payoff to a player is often taken to be the discounted sum of the stage payoffs or the limit inferior of the averages of the stage payoffs.

Stochastic games generalize both Markov decision processes and repeated games.

Stochastic two-player games on directed graphs are widely used for modeling and analysis of discrete systems operating in an unknown (adversarial) environment. Possible configurations of a system and its environment are represented as vertices, and the transitions correspond to actions of the system, its environment, or "nature". A run of the system then corresponds to an infinite path in the graph. Thus, a system and its environment can be seen as two players with antagonistic objectives, where one player (the system) aims at maximizing the probability of "good" runs, while the other player (the environment) aims at the opposite.

In many cases, there exists an equilibrium value of this probability, but optimal strategies for both players may not exist.

We introduce basic concepts and algorithmic questions studied in this area, and we mention some long-standing open problems. Then, we mention selected recent results.

The ingredients of a stochastic game are: a finite set of players ; a state space (either a finite set or a measurable space ); for each player , an action set (either a finite set or a measurable space ); a transition probability from , where is the action profiles, to , where is the probability that the next state is in given the current state and the current action profile ; and a payoff function from to , where the -th coordinate of , , is the payoff to player as a function of the state and the action profile .

The game starts at some initial state . At stage , players first observe , then simultaneously choose actions , then observe the action profile , and then nature selects according to the probability . A play of the stochastic game, , defines a stream of payoffs , where .

The discounted game with discount factor () is the game where the payoff to player is . The -stage game is the game where the payoff to player is .

The value , respectively , of a two-person zero-sum stochastic game , respectively , with finitely many states and actions exists, and Truman Bewley and Elon Kohlberg (1976) proved that converges to a limit as goes to infinity and that converges to the same limit as goes to .

The "undiscounted" game is the game where the payoff to player is the "limit" of the averages of the stage payoffs. Some precautions are needed in defining the value of a two-person zero-sum and in defining equilibrium payoffs of a non-zero-sum . The uniform value of a two-person zero-sum stochastic game exists if for every there is a positive integer and a strategy pair of player 1 and of player 2 such that for every and and every the expectation of with respect to the probability on plays defined by and is at least , and the expectation of with respect to the probability on plays defined by and is at most . Jean-François Mertens and Abraham Neyman (1981) proved that every two-person zero-sum stochastic game with finitely many states and actions has a uniform value.

If there is a finite number of players and the action sets and the set of states are finite, then a stochastic game with a finite number of stages always has a Nash equilibrium. The same is true for a game with infinitely many stages if the total payoff is the discounted sum.

The non-zero-sum stochastic game has a uniform equilibrium payoff if for every there is a positive integer and a strategy profile such that for every unilateral deviation by a player , i.e., a strategy profile with for all , and every the expectation of with respect to the probability on plays defined by is at least , and the expectation of with respect to the probability on plays defined by is at most . Nicolas Vieille has shown that all two-person stochastic games with finite state and action spaces have a uniform equilibrium payoff.

The non-zero-sum stochastic game has a limiting-average equilibrium payoff if for every there is a strategy profile such that for every unilateral deviation by a player , the expectation of the limit inferior of the averages of the stage payoffs with respect to the probability on plays defined by is at least , and the expectation of the limit superior of the averages of the stage payoffs with respect to the probability on plays defined by is at most . Jean-François Mertens and Abraham Neyman (1981) proves that every two-person zero-sum stochastic game with finitely many states and actions has a limiting-average value, and Nicolas Vieille has shown that all two-person stochastic games with finite state and action spaces have a limiting-average equilibrium payoff. In particular, these results imply that these games have a value and an approximate equilibrium payoff, called the liminf-average (respectively, the limsup-average) equilibrium payoff, when the total payoff is the limit inferior (or the limit superior) of the averages of the stage payoffs.

Whether every stochastic game with finitely many players, states, and actions, has a uniform equilibrium payoff, or a limiting-average equilibrium payoff, or even a liminf-average equilibrium payoff, is a challenging open question.

A Markov perfect equilibrium is a refinement of the concept of sub-game perfect Nash equilibrium to stochastic games.

Stochastic games have applications in economics, evolutionary biology and computer networks.^{[1]}^{[2]} They are generalizations of repeated games which correspond to the special case where there is only one state.

- ↑ Constrained Stochastic Games in Wireless Networks by E.Altman, K.Avratchenkov, N.Bonneau, M.Debbah, R.El-Azouzi, D.S.Menasche
- ↑ Djehiche, Boualem; Tcheukam, Alain; Tembine, Hamidou (2017-09-27). "Mean-Field-Type Games in Engineering".
*AIMS Electronics and Electrical Engineering*.**1**: 18–73. arXiv:1605.03281. doi:10.3934/ElectrEng.2017.1.18.

- Condon, A. (1992). "The complexity of stochastic games".
*Information and Computation*.**96**(2): 203–224. doi:10.1016/0890-5401(92)90048-K. - H. Everett (1957). "Recursive games". In Melvin Dresher, Albert William Tucker, Philip Wolfe (eds.).
*Contributions to the Theory of Games, Volume 3*. Annals of Mathematics Studies. Princeton University Press. pp. 67–78. ISBN 978-0-691-07936-3.CS1 maint: uses editors parameter (link) (Reprinted in Harold W. Kuhn, ed.*Classics in Game Theory*, Princeton University Press, 1997. ISBN 978-0-691-01192-9) - Filar, J. & Vrieze, K. (1997).
*Competitive Markov Decision Processes*. Springer-Verlag. ISBN 0-387-94805-8. - Mertens, J. F. & Neyman, A. (1981). "Stochastic Games".
*International Journal of Game Theory*.**10**(2): 53–66. doi:10.1007/BF01769259. - Neyman, A. & Sorin, S. (2003).
*Stochastic Games and Applications*. Dordrecht: Kluwer Academic Press. ISBN 1-4020-1492-9. - Shapley, L. S. (1953). "Stochastic games".
*PNAS*.**39**(10): 1095–1100. Bibcode:1953PNAS...39.1095S. doi:10.1073/pnas.39.10.1095. PMC 1063912. PMID 16589380. - Vieille, N. (2002). "Stochastic games: Recent results".
*Handbook of Game Theory*. Amsterdam: Elsevier Science. pp. 1833–1850. ISBN 0-444-88098-4. - Yoav Shoham; Kevin Leyton-Brown (2009).
*Multiagent systems: algorithmic, game-theoretic, and logical foundations*. Cambridge University Press. pp. 153–156. ISBN 978-0-521-89943-7. (suitable for undergraduates; main results, no proofs) - Tembine, H. Mean-field-type games. AIMS Mathematics, 2017, 2(4): 706-735. doi: 10.3934/Math.2017.4.706

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