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经济代写|微观经济学作业代写Microeconomics代考|Principles of classical game theory
Initiated by the founder book of von Neumann and Morgenstern (1944) and pursued by the pioneering work of Nash (1951), classical game theory is grounded on some simple interaction principles between players. The players match in either sequential or simultaneous meetings, define and implement some corresponding actions and get payoffs from the consequences of the combination of all actions. The players choose their actions in a strongly rational way, hence optimize their expected utility under various constraints, with regard to their beliefs about their material environment and their opponents. The players are coordinated in some equilibrium state, i. e. a stationary state (in absence of perturbations from outside the system) from which no player has an interest to deviate unilaterally. These three principles are examined successively.
In the first place, the game is structurally described by the opportunities of each player, the utility he gets from the conjunction of opportunities and the beliefs he holds about the preceding characteristics. In “normal form” (or “strategic form”), the game is formalized by a game matrix, which combines the possible actions of the players (and eventual states of nature) and expresses the utilities which are obtained for each combination of actions (and states). In “extensive form” (or “developed form”), the game is formalized by a game tree, which details the successive alternate moves of the players (and eventual states of nature) and indicates the utilities which are obtained for each path in the tree. The extensive form can be reduced to the normal form thanks to the notion of strategy, a strategy being defined by the action the player plays at each node where he has the move.
Then, each player is endowed with a strong rationality, both cognitive rationality in order to adapt his representations to his information and instrumental rationality in order to adapt his means to his objectives. Strong cognitive rationality implies that the player forms a perfect expectation of the future play of the game, by considering a correct specification of the game structure and a perfect and complete information on its past play. Strong instrumental rationality involves that the player choses an action by maximizing his expected utility (taking into account the random environment) under various individual and institutional constraints. Note that the same rationality applies to the actor for individual decision in a risky environment and in game theory, the “strategic uncertainty” about the opponent being in some sense “naturalized” in a “physical uncertainty”.
Finally, an equilibrium notion characterizes the social configurations in which the players’ actions are compatible, and leads to study the existence and multiplicity of the corresponding equilibrium states. In a normal-form game, a Nash equilibrium is a state where each action of a player is a best response to the others (equilibrium) actions, hence results from a fixed point of the best response functions. In an extensive-form game, a subgame perfect equilibrium is obtained by a backward induction procedure, each player playing at each time his best action, knowing the future best actions of all players. If these definitions explain the stability of some equilibrium state once established, they do not explain how an equilibrium state is achieved, except by introducing a fictitious entity, the Nash regulator, who appears as the precise counterpart in game theory of the Walrasian auctioneer.
经济代写|微观经济学作业代写Microeconomics代考|Limits of classical game theory
Classical game theory, in its original form, is based on very stringent assumptions as concerns the behavior and coordination of the players, but has tried to weaken them progressively. The players may be endowed with uncertain beliefs, either structural uncertainty on game rules and opponents’ (or nature’s) characteristics, or factual uncertainty on past opponents’ actions (or nature’s states) (Harsanyi, 1967). The players are endowed with bounded individual rationality, which constraints their reasoning abilities and leads them to chose suboptimal actions (Rubinstein, 1994). The players’ actions are no more coordinated towards an equilibrium state by the Nash regulator, but by crossed expectations on their actions and on their underlying determiners (Aumann, 1974). These limits will be examined successively.
First, classical game theory attempted to internalize the informational limits of the players by introducing sophisticated forms of uncertainty into the usual equilibrium notions. On the one hand, if he is not well aware of the opponents’ determiners assumed to be summarized in their types, a player gets endowed with a probability distribution on their possible types, eventually obtained from a common probability distribution. On the other hand, if he is not aware of his opponents’ past actions (symbolized by different nodes in the game tree), a player treats globally as an “information set” the nodes of the tree between which he cannot discriminate. More generally, it is usually assumed that any type of uncertainty can be reduced to a probabilistic form and even that this uncertainty is common knowledge among players (“each knows that the other knows that …”).
Second, classical game theory tried to internalize the computational limits of the players by introducing various forms of (cognitive) bounded rationality in the equilibrium notions. On the one hand, a player may be considered as acting as a finite state automaton, which is only able to implement strategies which are not too complex, especially strategies which only involve a bounded memory. On the other hand, a player is considered as a limited reasoning device, constrained by computing costs or astrained to finite internal states, which prevents him for instance to consider crossed beliefs above a given level. More generally, the players may still try to optimize their behavior on simplified decision problems or to treat the real decision problems with methods that are only approached ones.
Third, classical game theory gave “cognitive justifications” to its usual equilibrium functions, by assuming that the players are able, by their sole reasoning, to substitute for the Nash regulator. Contrary to weaker equilibrium notions, the justification of Nash equilibrium requires not only common knowledge of the game structure, of the players’ rationality and of the players’ independence of play and a common prior on actions, but even common knowledge of their respective conjectures, i. e. their expectations of others’ actions. The subgame perfect equilibrium is easier to justify, since it just needs a common knowledge of the game structure and players’ rationality, as well as of players’ independence of play. In fact, the players are again endowed with exceptional informational and computational capacities, not only in order to single out an equilibrium notion, but to select an equilibrium state in case of multiplicity.
经济代写|微观经济学作业代写MICROECONOMICS代考|Principles of evolutionist game theory
Evolutionist game theory, initiated twenty years ago (Maynard Smith, 1982) and still developing (Weibull, 1995; Young, 1998), aims at introducing soft assumptions concerning the adaptation of players along their dynamic interactions. The players have some reduced information on their opponents’ characteristics together with a limited spatially local and temporally bounded information on the past play of the game. The players are endowed with bounded rationality, their beliefs reducing at best to extrapolative expectations and their actions contenting with going in the right direction with regard to objectives. However, they are engaged in sequential encounters, in such a way that the repetitive work of time makes up for the role of the Nash regulator or of the crossed expectations. These three features will now be specified.
In the first place, evolutionist game theory considers the information gathered by a player as a by-product of his actions, considered as material rather than informational ones. Information concerns exceptionally some aspects of the game structure, either the utility (for the concerned player or his opponents) which may result from players’ conjoint actions or even the opponents’ prior beliefs. Information concerns more usually the sequence of actions implemented by the player’s opponents, at least in some “information neighborhood”, the player being assumed to know his own past actions. Information concerns finally the consequences of conjoint past actions, the utilities that the player gained by these consequences, and even the utilities that the other players got through their own actions.
As well, evolutionist game theory conceives the player’s rationality as essentially adaptive, in the sense that the player reacts to the past information by adjusting the parameters of a given choice rule. Cognitive rational-ity is grounded on a stationarity assumption as concerns player’s environment, the observed past actions or utilities being supposed to continue in average in the future. Instrumental rationality is grounded on an improving rather than an optimizing point of view, the choice rules being conceived in order to shift incrementally the action towards increasing utility. Moreover, the choice rules, now made duly explicit, incorporate the possibility of random deviations from their normal course, in order to explore from time to time the consequences of actions not spontaneously implemented.
Finally, evolutionist game theory postulates that the players, sometimes gathered in homogenous populations, meet systematically or randomly in a repeated and generally infinite sequence of repetitions of the same stage game. Each player meets only players situated in an “interaction neighborhood”, grounded on a spatial proximity criterion or a qualitative similarity criterion and possibly evolving endogenously over time. The players are engaged in a global dynamic process which, in some cases, converges asymptotically towards an equilibrium state, and gives a “evolutionist justification” to the associated equilibrium notion. Moreover, if the process converges, it leads (at least in probability when including random terms) towards some equilibrium state univocally defined by initial conditions, given context and past history, and gives therefore a constructive answer to the equilibrium multiplicity problem.
微观经济学代写
经济代写|微观经济学作业代写MICROECONOMICS代考|PRINCIPLES OF CLASSICAL GAME THEORY
由冯诺依曼和摩根斯坦的创始人书发起1944并被纳什的开创性工作所追求1951，经典博弈论是建立在玩家之间一些简单的交互原则之上的。玩家在连续或同时的会议中进行比赛，定义并实施一些相应的行动，并从所有行动组合的后果中获得回报。玩家以非常理性的方式选择他们的行动，从而优化他们在各种约束下的预期效用，关于他们对他们的物质环境和他们的对手的信念。参与者在某种平衡状态下进行协调，即静止状态一世n一种bs和nC和这Fp和r吨在rb一种吨一世这nsFr这米这在吨s一世d和吨H和s是s吨和米没有玩家有兴趣单方面偏离。这三个原则依次进行检验。
首先，博弈在结构上由每个参与者的机会、他从机会的结合中获得的效用以及他对先前特征的信念来描述。在“正常形式”这r“s吨r一种吨和G一世CF这r米”, 博弈由一个博弈矩阵形式化，它结合了玩家可能的行动一种nd和在和n吨在一种ls吨一种吨和s这Fn一种吨在r和并表示为每个动作组合获得的效用一种nds吨一种吨和s. 在“广泛的形式”这r“d和在和l这p和dF这r米”，博弈由博弈树形式化，详细描述了玩家的连续交替移动一种nd和在和n吨在一种ls吨一种吨和s这Fn一种吨在r和并指示为树中的每个路径获得的实用程序。由于策略的概念，扩展形式可以简化为正常形式，策略由玩家在他有移动的每个节点处执行的动作定义。
然后，每个参与者都被赋予了强大的理性，包括认知理性以使他的陈述适应他的信息和工具理性以使他的手段适应他的目标。强认知理性意味着玩家通过考虑对游戏结构的正确规范和关于其过去游戏的完美和完整的信息，形成了对游戏未来游戏的完美预期。强工具理性涉及玩家通过最大化其预期效用来选择行动吨一种ķ一世nG一世n吨这一种CC这在n吨吨H和r一种nd这米和n在一世r这n米和n吨在各种个人和制度限制下。请注意，同样的理性适用于在风险环境中和博弈论中进行个人决策的行为者，关于对手的“战略不确定性”在某种意义上被“归化”为“物理不确定性”。
最后，均衡概念刻画了参与者行为相容的社会配置，并导致研究相应均衡状态的存在性和多样性。在正常形式的博弈中，纳什均衡是一种状态，其中玩家的每个动作都是对其他人的最佳反应和q在一世l一世br一世在米动作，因此来自最佳响应函数的固定点。在广泛形式的博弈中，通过反向归纳过程获得子博弈完美均衡，每个玩家在每个时间玩他的最佳动作，知道所有玩家未来的最佳动作。如果这些定义解释了某些均衡状态一旦建立后的稳定性，那么它们并不能解释如何实现均衡状态，除非引入了一个虚构的实体，即纳什调节器，它在博弈论中与瓦尔拉斯拍卖师的精确对应物出现。
经济代写|微观经济学作业代写MICROECONOMICS代考|LIMITS OF CLASSICAL GAME THEORY
古典博弈论的原始形式基于非常严格的假设，涉及玩家的行为和协调，但试图逐渐削弱它们。玩家可能被赋予不确定的信念，要么是游戏规则的结构性不确定性，要么是对手的不确定性。这rn一种吨在r和′s过去对手行为的特征或事实不确定性这rn一种吨在r和′ss吨一种吨和s H一种rs一种n是一世,1967. 参与者被赋予有限的个体理性，这限制了他们的推理能力并导致他们选择次优行动R在b一世ns吨和一世n,1994. 参与者的行为不再由纳什监管者协调以达到平衡状态，而是通过对他们的行为及其潜在决定因素的交叉预期一种在米一种nn,1974. 这些限制将被依次检查。
首先，经典博弈论试图通过将复杂形式的不确定性引入通常的均衡概念来内化参与者的信息限制。一方面，如果他不太了解假定的对手类型的决定因素，那么玩家就会被赋予其可能类型的概率分布，最终从一个共同的概率分布中获得。另一方面，如果他不知道对手过去的行为s是米b这l一世和和db是d一世FF和r和n吨n这d和s一世n吨H和G一种米和吨r和和，玩家将全局视为“信息集”，即他无法区分的树的节点。更一般地说，通常假设任何类型的不确定性都可以简化为概率形式，甚至这种不确定性是参与者之间的常识“和一种CHķn这在s吨H一种吨吨H和这吨H和rķn这在s吨H一种吨…”.
其次，经典博弈论试图通过引入各种形式的C这Gn一世吨一世在和均衡概念中的有限理性。一方面，玩家可以被认为是一个有限状态自动机，它只能执行不太复杂的策略，尤其是只涉及有界记忆的策略。另一方面，玩家被认为是有限的推理设备，受到计算成本的限制或受限于有限的内部状态，这会阻止他考虑例如超过给定水平的交叉信念。更一般地说，玩家可能仍然会尝试优化他们在简化决策问题上的行为，或者使用仅接近的方法来处理真正的决策问题。
第三，经典博弈论为其通常的均衡函数提供了“认知理由”，假设参与者能够通过他们唯一的推理来替代纳什调节器。与较弱的均衡概念相反，纳什均衡的证明不仅需要对博弈结构、参与者的理性和参与者的游戏独立性和行动的共同先验的常识，甚至对他们各自猜想的常识，即他们对他人行为的期望。子博弈完美均衡更容易证明，因为它只需要对博弈结构和博弈者的理性以及博弈者的独立性有一个共同的认识。事实上，玩家再次被赋予了非凡的信息和计算能力，
经济代写|微观经济学作业代写MICROECONOMICS代考|PRINCIPLES OF EVOLUTIONIST GAME THEORY
进化论博弈论，二十年前发起米一种是n一种rd小号米一世吨H,1982并且仍在发展中在和一世b在ll,1995;是这在nG,1998，旨在引入有关玩家在动态交互中的适应的软假设。玩家对他们的对手的特征有一些减少的信息，以及关于过去比赛的有限的空间局部和时间限制信息。参与者被赋予有限理性，他们的信念充其量减少到外推期望，他们的行动满足于朝着正确的目标方向前进。然而，他们参与了连续的遭遇，以这样一种方式，时间的重复工作弥补了纳什监管者或交叉预期的作用。现在将指定这三个特征。
首先，进化论博弈论将玩家收集的信息视为其行为的副产品，被视为物质而非信息。信息特别关注游戏结构的某些方面，无论是效用F这r吨H和C这nC和rn和dpl一种是和r这rH一世s这pp这n和n吨s这可能是由于玩家的联合行动，甚至是对手的先前信念。信息通常涉及玩家的对手执行的动作序列，至少在某些“信息邻域”中，假设玩家知道他自己过去的动作。信息最终涉及联合过去行动的后果，参与者通过这些后果获得的效用，甚至其他参与者通过自己的行动获得的效用。
同样，进化论博弈论认为玩家的理性本质上是适应性的，即玩家通过调整给定选择规则的参数来对过去的信息做出反应。认知理性基于关于玩家环境的平稳性假设，观察到的过去行为或效用应该在未来平均继续。工具理性基于改进而非优化的观点，选择规则的构思是为了逐步将行动转向增加效用。此外，现在已正式明确的选择规则包含了随机偏离其正常过程的可能性，以便不时探索未自发实施的行动的后果。
最后，进化论博弈论假设玩家，有时聚集在同质人群中，在同一阶段游戏的重复且通常无限的重复序列中系统地或随机地相遇。每个玩家只遇到位于“交互邻域”中的玩家，基于空间接近度标准或定性相似度标准，并且可能随着时间的推移内生发展。参与者参与了一个全局动态过程，在某些情况下，该过程会渐近趋于平衡状态，并为相关的平衡概念提供“进化论理由”。此外，如果过程收敛，它会导致一种吨l和一种s吨一世npr这b一种b一世l一世吨是在H和n一世nCl在d一世nGr一种nd这米吨和r米s朝着由初始条件、给定背景和过去历史明确定义的某种均衡状态，因此对均衡多重性问题给出了建设性的答案。
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