game theory.docx

What is GAME THEORY?Game theoryis the study of strategicdecision making. Specifically, it is the study ofmathematical modelsof conflict and cooperation between intelligent rational decision-makers.1An alternative term suggested as a more descriptive name for the discipline isinteractivedecision theory.2The subject first addressedzero-sum games, such that one persons gains exactly equal net losses of the other participant or participants.Game theory is mainly used in economics, political science, and psychology, as well as logic, computer science, and biology. Today, however, game theory applies to a wide range of behavioral relations, and has developed into anumbrella termfor the logical side of decision science, including both humans and non-humans.Elements: theplayersof the game, theinformationandactionsavailable to each player at each decision point, and thepayoffsfor each outcome.Classification纳什均衡(Nash equilibrium），Ingame theory, theNash equilibriumis asolution conceptof anon-cooperative gameinvolving two or more players, in which each player is assumed to know the equilibrium strategies of the other players, and no player has anything to gain by changing only their own strategy.1If each player has chosen a strategy and no player can benefit by changing strategies while the other players keep theirs unchanged, then the current set of strategy choices and the corresponding payoffs constitutes a Nash equilibrium. The reality of the Nash equilibrium of a game can be tested usingexperimental economicsmethod.Stated simply, Amy and Will are in Nash equilibrium if Amy is making the best decision she can, taking into account Wills decision while Wills decision remains unchanged, and Will is making the best decision he can, taking into account Amys decision while Amys decision remains unchanged. Likewise, a group of players are in Nash equilibrium if each one is making the best decision possible, taking into account the decisions of the others in the game as long the other partys decision remains unchanged.纳什均衡是一种策略组合，使得每个参与人的策略是对其他参与人策略的最优反应。假设有n个局中人参与博弈，如果某情况下无一参与者可以独自行动而增加收益（即为了自身利益的最大化，没有任何单独的一方愿意改变其策略的1），则此策略组合被称为纳什均衡。所有局中人策略构成一个策略组合（Strategy Profile）。纳什均衡，从实质上说，是一种非合作博弈状态。纳什均衡达成时，并不意味着博弈双方都处于不动的状态，在顺序博弈中这个均衡是在博弈者连续的动作与反应中达成的。纳什均衡也不意味着博弈双方达到了一个整体的最优状态，需要注意的是，只有最优策略才可以达成纳什均衡，严格劣势策略不可能成为最佳对策，而弱优势和弱劣势策略是有可能达成纳什均衡的。在一个博弈中可能有一个以上的纳什均衡，而囚徒困境中有且只有一个纳什均衡。子博弈精炼纳什均衡（subgame perfect Nash equilibrium），贝叶斯纳什均衡(Bayesian Nash equilibrium），精炼贝叶斯纳什均衡(perfect Bayesian Nash equilibrium）Cooperative / Non-cooperativeeditMain articles:Cooperative gameandNon-cooperative gameA game iscooperativeif the players are able to form binding commitments. For instance, the legal system requires them to adhere to their promises. In noncooperative games, this is not possible.Often it is assumed thatcommunicationamong players is allowed in cooperative games, but not in non-cooperative ones. However, this classification on two binary criteria has been questioned, and sometimes rejected.41Of the two types of games, noncooperative games are able to model situations to the finest details, producing accurate results. Cooperative games focus on the game at large. Considerable efforts have been made to link the two approaches. The so-called Nash-programme (Nash program is the research agenda for investigating on the one hand axiomatic bargaining solutions and on the other hand the equilibrium outcomes of strategic bargaining procedures)42has already established many of the cooperative solutions as noncooperative equilibria.Hybridgames contain cooperative and non-cooperative elements. For instance, coalitions of players are formed in acooperative game, but these play in a non-cooperative fashion.Zero-sum / Non-zero-sumeditABA1, 13, 3B0, 02, 2A zero-sum gameMain article:Zerosum gameZero-sum games are a special case of constant-sum games, in which choices by players can neither increase nor decrease the available resources. In zero-sum games the total benefit to all players in the game, for every combination of strategies, always adds to zero (more informally, a player benefits only at the equal expense of others).Pokerexemplifies a zero-sum game (ignoring the possibility of the houses cut), because one wins exactly the amount ones opponents lose. Other zero-sum games includematching penniesand most classical board games includingGoandchess.Many games studied by game theorists (including the infamousprisoners dilemma) are non-zero-sum games, because theoutcomehas net results greater or less than zero. Informally, in non-zero-sum games, a gain by one player does not necessarily correspond with a loss by another.Constant-sum games correspond to activities like theft and gambling, but not to the fundamental economic situation in which there are potentialgains from trade. It is possible to transform any game into a (possibly asymmetric) zero-sum game by adding a dummy player (often called the board) whose losses compensate the players net winnings.Founder John Forbes Nash, Jr.(June 13, 1928 May 23, 2015) was an Americanmathematicianwhose works ingame theory,differential geometry, andpartial differential equationshave provided insight into the factors that govern chance and events inside complex systems in daily life.Game theoryNash earned aPh.D.degree in 1950 with a 28-page dissertation onnon-cooperative games Princeton college.910The thesis, which was written under the supervision of doctoral advisorAlbert W. Tucker, contained the definition and properties of the Nash equilibrium. A crucial concept in non-cooperative games, it won Nash theNobel Memorial Prize in Economic Sciencesin 1994.he shared the 1994Nobel Memorial Prize in Economic Scienceswith game theoristsReinhard SeltenandJohn Harsanyi. In 2015, he was awarded theAbel Prizefor his work onnonlinearpartial differential equations.In 1959, Nash began showing clear signs of mental illness, and spent several years at psychiatric hospitals being treated forparanoid schizophrenia. After 1970, his condition slowly improved, allowing him to return to academic work by the mid-1980s.He went to Princeton, where he worked on his equilibrium theory, later known as theNash equilibrium.Case one: “囚徒困境”是1950年美国兰德公司提出的博弈论模型。两个共谋犯罪的人被关入监狱，不能互相沟通情况。如果两个人都不揭发对方，则由于证据不确定，每个人都坐牢一年；若一人揭发，而另一人沉默，则揭发者因为立功而立即获释，沉默者因不合作而入狱五年；若互相揭发，则因证据确实，二者都判刑两年。由于囚徒无法信任对方，因此倾向于互相揭发，而不是同守沉默。囚徒困境（prisoners dilemma ）：两个被捕的囚徒之间的一种特殊博弈，说明为什么甚至在合作对双方都有利时，保持合作也是困难的。囚徒困境是博弈论的非零和博弈中具代表性的例子，反映个人最佳选择并非团体最佳选择。虽然困境本身只属模型性质，但现实中的价格竞争、环境保护等方面，也会频繁出现类似情况。囚徒困境的故事讲的是，两个嫌疑犯作案后被警察抓住，分别关在不同的屋子里接受审讯。警察知道两人有罪，但缺乏足够的证据。警察告诉每个人：如果两人都抵赖，各判刑一年；如果两人都坦白，各判八年；如果两人中一个坦白而另一个抵赖，坦白的放出去，抵赖的判十年。于是，每个囚徒都面临两种选择：坦白或抵赖。然而，不管同伙选择什么，每个囚徒的最优选择是坦白：如果同伙抵赖、自己坦白的话放出去，抵赖的话判十年，坦白比不坦白好；如果同伙坦白、自己坦白的话判八年，比起抵赖的判十年，坦白还是比抵赖的好。结果，两个嫌疑犯都选择坦白，各判刑八年。如果两人都抵赖，各判一年，显然这个结果好。Theprisoners dilemmais a canonical example of a game analyzed ingame theorythat shows why two purely rational individuals might not cooperate, even if it appears that it is in their best interestscitation neededto do so. It was originally framed byMerrill FloodandMelvin Dresherworking atRANDin 1950.Albert W. Tuckerformalized the game with prison sentence rewards and gave it the name prisoners dilemma (Poundstone, 1992), presenting it as follows:Two members of a criminal gang are arrested and imprisoned. Each prisoner is in solitary confinement with no means of speaking to or exchanging messages with the other. The prosecutors do not have enough evidence to convict the pair on the principal charge. They hope to get both sentenced to a year in prison on a lesser charge. Simultaneously, the prosecutors offer each prisoner a bargain. Each prisoner is given the opportunity either to: betray the other by testifying that the other committed the crime, or to cooperate with the other by remaining silent. Here is the offer: If A and B each betray the other, each of them serves 2 years in prison If A betrays B but B remains silent, A will be set free and B will serve 3 years in prison (and vice versa) If A and B both remain silent, both of them will only serve 1 year in prison (on the lesser charge)It is implied that the prisoners will have no opportunity to reward or punish their partner other than the prison sentences they get, and that their decision will not affect their reputation in the future. Because betraying a partner offers a greater reward than cooperating with him, all purely rational self-interested prisoners would betray the other, and so the only possible outcome for two purely rational prisoners is for them to betray each other.1The interesting part of this result is that pursuing individual reward logically leads both of the prisoners to betray, when they would get a