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经济代写|博弈论代写Game Theory代考|Interaction, Negotiation, and Learning

Most of the two-player games in Chapter 3 have simultaneous moves, where both players choose their actions without knowing the action of the partner and do not change their choice once the action of the partner becomes known. These are referred to as one-shot, simultaneous, or sealed-bid games. We have already argued that real interactions are often not of this form (e.g. Section 3.5). Instead there is a sequence of decisions where each decision is contingent on the current information, and in particular on what has gone before. In this interaction over time there is often learning about the partner or the situation.

For example, consider the decision of parents whether to care for their young or to desert. McNamara et al. (2002) review evidence that the decisions of the two parents are often not independent. One of the cases they consider is that of the penduline tit. In the data reproduced in Fig. 8.1, out of $n=130$ clutches the proportion in which the male cares is $p_m=(0+25) / 130=0.19$. Similarly the proportion in which the female cares is $p_f=(0+67) / 130=0.52$. If individuals are following a mixed-strategy Nash equilibrium with independent choices, then from these marginal proportions an estimate of the number of clutches in which there is biparental care is $n_{\text {bipar }}=$ $n p_m p_f=130 \times(25 / 130) \times(67 / 130)=12.9$. A comparison of observed values with those expected under independence provides strong evidence that decisions are not independent (the null hypothesis of independence can be rejected at $p<0.0001$ in a chi-squared test). It is striking that there are no cases of biparental care. The obvious inference is that if one parent is caring the other notices this and leaves.

This chapter is concerned with the interaction process. By process we mean the sequence of actions and the information on which each action is based. As we show, one cannot predict the outcome of an interaction from the payoffs alone, but one must know the process. For example, just changing the order in which players choose their actions can completely change predictions (Section 8.2). Changing the order of choice has the effect of changing the information available to players. An individual can gain an advantage by giving reliable information to a partner. This might be achieved by individuals committing themselves to a course of action by deliberately handicapping themselves (Section 8.3), or through reputation effects. In an interaction individuals may differ in their abilities, energy reserves, or other aspects of state. Often the state of an individual is private information to that individual that is partly revealed by its actions. The interaction may then be a process in which the individuals gain partial information about one another, although it may be in the interests of each to hide information. So for example, it can be advantageous to a parent to appear less able to care than is actually the case, to force the mate to increase its care effort in order to compensate (Sections $8.4$ and 8.5). It may even be the case that individuals do not know their own state and must learn about that, for instance when establishing dominance relations in a social group. This and other learning scenarios are considered in Sections 8.5, 8.6, and 8.7. Learning mechanisms can be especially helpful to study interactions over time when individuals respond to each other’s actions that in turn are influenced by individual characteristics. Such a large-worlds perspective (Chapter 5) could also be realistic in terms of the traits and mechanisms assumed. We end the chapter (Section 8.8) with a general discussion of game theory for interaction over time, including the role of behavioural mechanisms and how one can think about evolutionary stability for large-worlds models.

经济代写|博弈论代写Game Theory代考|Information and the Order of Choice

Consider two parents that have produced young together. Suppose that each can either care for the young (C) or desert (D). The advantage of caring is that the young do better, which benefits both parents. However, care is costly and reduces prospects for remating. For illustrative purposes, let the payoffs to the two parents be given by Table 8.1. This payoff structure might be appropriate if the male pays a greater cost for uniparental care than the female and also gains less from desertion, perhaps because the local population is male biased. We compare two processes by which the actions are chosen. Suppose first that each parent chooses its action without knowledge of the decision of the other parent. Each then sticks to its decision once the decision of the other becomes known. In game theory this scenario is known as simultaneous moves. From Table $8.1$ it can be seen that whatever the action of the female the male does best if he cares. Given that the male will care the female’s best action is to desert. Thus the only Nash equilibrium is for the male to always care and the female to always desert.
Now suppose that the male chooses his action first. The female then chooses her action knowing the choice of the male. A game with this information structure, where one player makes its decision knowing that of the partner, is known as a Stackelberg game. We can represent this scenario by the game tree in Fig. 8.2. Note that the payoff structure is exactly as in Table 8.1. We can find a Nash equilibrium by working backwards. If the male cares the female will desert and the male will get payoff 3 . If the male deserts the female will care and the male will get payoff 4 . Thus the male should desert and the female will then care. The Nash equilibrium found by working backwards in this way is called a subgame perfect equilibrium (see Section 8.3).
If we compare the outcomes of the two processes we see they predict opposite forms of care. In the simultaneous choice case the male cares, in the sequential choice case he deserts, despite the fact that his best action is to care for either of the fixed actions of the female. This example shows that, by itself, the payoff structure is not enough to predict evolutionary outcomes; we must also specify the decision process. The evidence in Fig. $8.1$ suggests that the penduline tits do not make simultaneous decisions. The process is not, however, liable to be as simple as one of the parents choosing first and the other reacting to this choice (Griggio et al., 2004; van Dijk et al., 2007). For example, during foraging trips each may assess the possibilities to remate if they desert, so that actions may also be contingent on this information. Furthermore, there is evidence that the female attempts to hide the fact she has already produced eggs from the male (Valera et al., 1997). This may allow her to gain an advantage by being the first to choose.

经济代写|博弈论代写Game Theory代考|ECON90022

博弈论代考

经济代写|博弈论代写博弈论代考|互动、谈判和学习

.


第三章中的大多数双人游戏都有同步的行动,即双方玩家在不知道同伴的行动的情况下选择自己的行动,并且在知道同伴的行动后也不会改变自己的选择。这些被称为一次性、同时或密封投标的游戏。我们已经讨论过,真正的交互通常不是这种形式(如第3.5节)。取而代之的是一系列的决定,每个决定都取决于当前的信息,特别是之前发生的事情。随着时间的推移,在这种互动中,人们通常会了解对方或当时的情况


例如,考虑父母的决定是照顾他们的孩子还是抛弃他们。McNamara等人(2002)回顾了父母双方的决定往往不是独立的证据。他们考虑的一个例子是摆山雀。在图8.1中复制的数据中,在$n=130$离合器中,雄性关心的比例是$p_m=(0+25) / 130=0.19$。同样,女性关心的比例是$p_f=(0+67) / 130=0.52$。如果个体遵循一种具有独立选择的混合策略纳什均衡,那么从这些边际比例可以估计出双亲照顾的群体数量为$n_{\text {bipar }}=$$n p_m p_f=130 \times(25 / 130) \times(67 / 130)=12.9$。观察值与独立性下预期值的比较提供了强有力的证据,表明决策不是独立的(独立的零假设可以在$p<0.0001$上的卡方检验中被拒绝)。令人惊讶的是,这里没有双亲照顾的案例。显而易见的推论是,如果父母中的一方关心孩子,另一方就会注意到并离开

本章涉及交互过程。所谓过程,我们指的是行动的顺序和每个行动所基于的信息。正如我们所展示的,人们不能仅从收益来预测互动的结果,但必须知道这个过程。例如,仅仅改变玩家选择行动的顺序就可以完全改变预测。改变选择顺序将改变玩家所能获得的信息。个人可以通过向伙伴提供可靠的信息而获得优势。这可以通过个人通过故意妨碍自己(第8.3节)或通过声誉效应来实现。在相互作用中,个体可能在能力、能量储备或状态的其他方面有所不同。通常情况下,一个人的状态对这个人来说是私人信息,部分是通过他的行为暴露出来的。这种互动可能是一个过程,在这个过程中,个体之间获得了关于彼此的部分信息,尽管隐藏信息可能符合双方的利益。因此,举例来说,如果父母表现得比实际情况更缺乏照顾能力,迫使配偶增加照顾努力,以补偿($8.4$节和8.5节),这可能是有利的。甚至可能出现这样的情况:个体不知道自己的状态,必须了解这一点,例如在一个社会群体中建立支配关系时。这和其他学习场景将在第8.5、8.6和8.7节中讨论。随着时间的推移,当个体对彼此行为做出反应时,学习机制对研究相互作用特别有帮助,而这些行为又受到个体特征的影响。这样的大世界视角(第5章)在假设的特征和机制方面也可能是现实的。在本章(第8.8节)的最后,我们将对长期互动的博弈论进行一般性讨论,包括行为机制的作用,以及人们如何思考大世界模型的进化稳定性

经济代写|博弈论代写博弈论代考|信息和选择的顺序


试想一对共同生育后代的父母。假设双方都能照顾幼崽(C)或抛弃幼崽(D)。照顾幼崽的好处是幼崽做得更好,这对父母双方都有利。然而,护理是昂贵的,并降低了重新分娩的可能性。为了便于说明,让表8.1给出对两个父母的收益。这种回报结构可能是合适的,如果雄性为单亲抚养付出的代价比雌性大,而且从遗弃中获得的收益也更少,可能是因为当地人口对雄性的偏见。我们比较了选择动作的两个过程。首先假设父母双方都在不知道另一方的决定的情况下选择自己的行为。一旦对方的决定被知晓,双方就会坚持自己的决定。在博弈论中,这种情况被称为同时走法。从表$8.1$可以看出,无论女性的行动,如果男性关心,他做得最好。考虑到雄性会关心雌性,最好的做法就是逃跑。因此,唯一的纳什均衡是男性总是关心,而女性总是抛弃。现在假设雄性先选择行动。在了解雄性的选择后,雌性会选择自己的行为。具有这种信息结构的游戏,即一个玩家在知道同伴的决定的情况下做出决定,被称为Stackelberg游戏。我们可以用图8.2中的游戏树来表示这个场景。注意,收益结构完全如表8.1所示。我们可以通过逆向计算找到纳什均衡。如果雄性关心,雌性就会放弃,雄性就会得到回报。如果雄性抛弃了雌性,雌性会在意,雄性会得到回报。因此,雄性应该放弃,而雌性则会关心。通过这种方法找到的纳什均衡被称为子博弈完美均衡(见8.3节)。如果我们比较这两个过程的结果,我们会发现它们预测的是相反的护理形式。在同步选择的情况下,男性会关心,而在顺序选择的情况下,他会放弃,尽管他的最佳行动是关心女性的固定行动中的任何一个。这个例子表明,收益结构本身不足以预测进化结果;我们还必须指定决策过程。图$8.1$中的证据表明,钟摆山雀不会同时做出决定。然而,这个过程并不像父母中的一方先做出选择,另一方对这个选择做出反应那样简单(Griggio等人,2004年;van Dijk et al., 2007)。例如,在觅食之旅中,每个人都可能评估如果他们逃跑的可能性,所以行动也可能取决于这个信息。此外,有证据表明,雌性试图隐瞒她已经为雄性产卵的事实(Valera et al., 1997)。这可能使她通过成为第一个做出选择的人而获得优势

经济代写|博弈论代写Game Theory代考

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