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 **solution concept** is a formal rule for predicting how a game will be played. These predictions are called "solutions", and describe which strategies will be adopted by players and, therefore, the result of the game. The most commonly used solution concepts are equilibrium concepts, most famously Nash equilibrium.

Many solution concepts, for many games, will result in more than one solution. This puts any one of the solutions in doubt, so a game theorist may apply a **refinement** to narrow down the solutions. Each successive solution concept presented in the following improves on its predecessor by eliminating implausible equilibria in richer games.

Let be the class of all games and, for each game , let be the set of strategy profiles of . A *solution concept* is an element of the direct product *i.e*., a function such that for all

In this solution concept, players are assumed to be rational and so **strictly dominated strategies** are eliminated from the set of strategies that might feasibly be played. A strategy is strictly dominated when there is some other strategy available to the player that always has a higher payoff, regardless of the strategies that the other players choose. (Strictly dominated strategies are also important in minimax game-tree search.) For example, in the (single period) prisoners' dilemma (shown below), *cooperate* is strictly dominated by *defect* for both players because either player is always better off playing *defect*, regardless of what his opponent does.

Prisoner 2 Cooperate | Prisoner 2 Defect | |
---|---|---|

Prisoner 1 Cooperate | −0.5, −0.5 | −10, 0 |

Prisoner 1 Defect | 0, −10 | −2, −2 |

A Nash equilibrium is a strategy profile (a strategy profile specifies a strategy for every player, e.g. in the above prisoners' dilemma game (*cooperate*, *defect*) specifies that prisoner 1 plays *cooperate* and player 2 plays *defect*) in which every strategy is a best response to every other strategy played. A strategy by a player is a best response to another player's strategy if there is no other strategy that could be played that would yield a higher pay-off in any situation in which the other player's strategy is played.

There are games that have multiple Nash equilibria, some of which are unrealistic. In the case of dynamic games, unrealistic Nash equilibria might be eliminated by applying backward induction, which assumes that future play will be rational. It therefore eliminates noncredible threats because such threats would be irrational to carry out if a player was ever called upon to do so.

For example, consider a dynamic game in which the players are an incumbent firm in an industry and a potential entrant to that industry. As it stands, the incumbent has a monopoly over the industry and does not want to lose some of its market share to the entrant. If the entrant chooses not to enter, the payoff to the incumbent is high (it maintains its monopoly) and the entrant neither loses nor gains (its payoff is zero). If the entrant enters, the incumbent can fight or accommodate the entrant. It will fight by lowering its price, running the entrant out of business (and incurring exit costs – a negative payoff) and damaging its own profits. If it accommodates the entrant it will lose some of its sales, but a high price will be maintained and it will receive greater profits than by lowering its price (but lower than monopoly profits).

If the entrant enters, the best response of the incumbent is to accommodate. If the incumbent accommodates, the best response of the entrant is to enter (and gain profit). Hence the strategy profile in which the incumbent accommodates if the entrant enters and the entrant enters if the incumbent accommodates is a Nash equilibrium. However, if the incumbent is going to play fight, the best response of the entrant is to not enter. If the entrant does not enter, it does not matter what the incumbent chooses to do (since there is no other firm to do it to - note that if the entrant does not enter, fight and accommodate yield the same payoffs to both players; the incumbent will not lower its prices if the entrant does not enter). Hence fight can be considered as a best response of the incumbent if the entrant does not enter. Hence the strategy profile in which the incumbent fights if the entrant does not enter and the entrant does not enter if the incumbent fights is a Nash equilibrium. Since the game is dynamic, any claim by the incumbent that it will fight is an incredible threat because by the time the decision node is reached where it can decide to fight (i.e. the entrant has entered), it would be irrational to do so. Therefore, this Nash equilibrium can be eliminated by backward induction.

See also:

A generalization of backward induction is subgame perfection. Backward induction assumes that all future play will be rational. In subgame perfect equilibria, play in every subgame is rational (specifically a Nash equilibrium). Backward induction can only be used in terminating (finite) games of definite length and cannot be applied to games with imperfect information. In these cases, subgame perfection can be used. The eliminated Nash equilibrium described above is subgame imperfect because it is not a Nash equilibrium of the subgame that starts at the node reached once the entrant has entered.

Sometimes subgame perfection does not impose a large enough restriction on unreasonable outcomes. For example, since subgames cannot cut through information sets, a game of imperfect information may have only one subgame – itself – and hence subgame perfection cannot be used to eliminate any Nash equilibria. A perfect Bayesian equilibrium (PBE) is a specification of players’ strategies *and beliefs* about which node in the information set has been reached by the play of the game. A belief about a decision node is the probability that a particular player thinks that node is or will be in play (on the *equilibrium path*). In particular, the intuition of PBE is that it specifies player strategies that are rational given the player beliefs it specifies and the beliefs it specifies are consistent with the strategies it specifies.

In a Bayesian game a strategy determines what a player plays at every information set controlled by that player. The requirement that beliefs are consistent with strategies is something not specified by subgame perfection. Hence, PBE is a consistency condition on players’ beliefs. Just as in a Nash equilibrium no player's strategy is strictly dominated, in a PBE, for any information set no player's strategy is strictly dominated beginning at that information set. That is, for every belief that the player could hold at that information set there is no strategy that yields a greater expected payoff for that player. Unlike the above solution concepts, no player's strategy is strictly dominated beginning at any information set even if it is off the equilibrium path. Thus in PBE, players cannot threaten to play strategies that are strictly dominated beginning at any information set off the equilibrium path.

The *Bayesian* in the name of this solution concept alludes to the fact that players update their beliefs according to Bayes' theorem. They calculate probabilities given what has already taken place in the game.

**Forward induction** is so called because just as backward induction assumes future play will be rational, forward induction assumes past play was rational. Where a player does not know what *type* another player is (i.e. there is imperfect and asymmetric information), that player may form a belief of what type that player is by observing that player's past actions. Hence the belief formed by that player of what the probability of the opponent being a certain type is based on the past play of that opponent being rational. A player may elect to signal his type through his actions.

Kohlberg and Mertens (1986) introduced the solution concept of Stable equilibrium, a refinement that satisfies forward induction. A counter-example was found where such a stable equilibrium did not satisfy backward induction. To resolve the problem Jean-François Mertens introduced what game theorists now call Mertens-stable equilibrium concept, probably the first solution concept satisfying both forward and backward induction.

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