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经济代写|博弈论代写Game Theory代考|A Model of the Nectar Market

Assume that a plant species relies on a particular insect for pollination. Visits by insects are rare and between visits a plant builds up the amount of nectar present in a flower until it is at some genetically determined level $x$. When an insect visits a flower it consumes nectar; its rate of consumption is $r(z)$ when the current amount of nectar in the flower is $z$. The rate of change in the amount of nectar during a visit is thus $\frac{d z}{d t}=-r(z)$. We assume that $r(z)$ is an increasing function of $z$, so that nectar is harder to extract as its level falls. The insect leaves when its instantaneous consumption rate falls to $r(z)=\gamma$, where $\gamma$ can depend on the availability of nectar in the environment as a whole. The insect then moves on to another plant and again forages for nectar. The mean time taken to find another flower after leaving the previous flower is $\tau$.

We assume that insects choose their leaving rate $\gamma$ so as to maximize the average rate at which they obtain nectar. Plants gain a benefit $B(T)$ if the duration of an insect’s visit is $T$. This is an increasing but decelerating function of the visit duration. They also pay a cost per visit equal to the amount of nectar consumed-this amount is equal to the amount of additional nectar that the plant must produce in order to replenish the amount present to its original level. The payoff to a plant equals the benefit minus cost.
To analyse this model assume that the resident strategy is for plants to provision flowers to level $x$. Let $z_\gamma$ satisfy $r\left(z_\gamma\right)=\gamma$, so that $z_\gamma$, is the amount of nectar left when an insects leaves a flower. Then the insect gains $x-z_\gamma$ nectar on each flower. Let $T(x, z)$ be the time taken to deplete a flower from its initial nectar level of $x$ to level $z$. The mean rate at which the insect consumes nectar is $g(x, \gamma)=\frac{x-z_\gamma}{T\left(x, z_\gamma\right)+\tau}$. This mean rate is maximized by setting $\gamma=\gamma^(x)$ where $\gamma^(x)$ is the unique solution of the equation $g\left(x, \gamma^(x)\right)=\gamma^(x)$ (McNamara, 1985). Thus insects leave a flower when their instantaneous consumption rate falls to $\gamma^(x)$, at which time the amount of nectar left in a flower is $z_{\gamma^(x)}$.

Now consider a rare mutant plant that provisions to level $x^{\prime}$ when the resident provisioning level is $x$. An insect stays for time $T\left(x^{\prime}, z_{\gamma^{+}(x)}\right)$ on a mutant flower. The payoff to the mutant is thus
$$
W\left(x^{\prime}, x\right)=B\left(T\left(x^{\prime}, z_{\gamma^(x)}\right)\right)-\left(x^{\prime}-z_{\gamma^(x)}\right) .
$$
Figure $7.9$ illustrates the effect of the resident strategy on the mean rate at which insects gain nectar and the best response of a rare mutant plant. As the resident provisioning level $x$ increases an insect encounters more nectar on each flower and its instantaneous rate of consumption on arriving at a flower increases.

经济代写|博弈论代写Game Theory代考|Choosiness, Assortment, and Cooperation

We now consider markets in which there is just a single species, with population members acting both as service providers and recipients of service. Specifically, we assume that individuals pair up to perform a task that has potential benefits to the pair members, but where contributing to the common good has costs to the individual (cf. Section 3.1). Assume that there is variation in the level of contribution of pair members. We outline how the existence of variation in a market setting can lead to a positive correlation in the contribution of pair members. We then argue that the establishment of such a correlation selects for a greater contribution to the common good; i.e. greater cooperation between pair members.

In a market setting there are two main ways in which choosiness about a partner can lead to a positive correlation in the level of contribution of pair members. Both mechanisms rely on the fact that a pair forms or stays together if and only if both partners decide to do so.

  1. Before a pair is formed individuals may have information about the likely contribution of prospective partners. This information might, for example, be as a result of reputation based on previous behaviour with others. Suppose also that in seeking a partner, individuals are able to reject one individual in favour of a more cooperative alternative. Then population members that are highly cooperative will tend to be rejected by few potential partners and so can afford to be choosy, whereas individuals that are uncooperative will be rejected by many and cannot afford to be choosy. As a consequence, highly cooperative individuals will tend to pair up with other highly cooperative individuals, leading to a positive correlation in the cooperativeness of members of a pair. McNamara and Collins (1990) illustrate this effect in the context of mutual mate choice.
  2. Individuals may be able to break off their interaction with an uncooperative partner and seek another, more cooperative, partner. Then pairs will tend to stay together if both pair members are highly cooperative. In contrast, if a highly cooperative individual has an uncooperative partner it will ‘divorce’ the partner and seek a better one. As a result, there is again a positive correlation in cooperativeness. McNamara et al. (1999a) illustrate this effect in the context of mating and divorce by both sexes.

Other processes have been suggested that could potentially lead to a positive assortment between non-relatives (e.g. Joshi et al., 2017).

The existence of a positive correlation exerts a selection pressure on the trait that is correlated. Box $7.4$ presents an example in which, regardless of the source of the correlation, increased correlation leads to increased cooperation at evolutionary stability. Two cases are considered. In the first the variation in the contribution to the common good might be thought of as arising through developmental noise. In the second, there is no developmental noise but processes such as mutation maintain genetic variation. Both cases predict the same effect.

经济代写|博弈论代写Game Theory代考|ECOS3012

博弈论代考

经济代写|博弈论代写博弈论代考|甘露市场的一个模型


假设一种植物依赖一种特定的昆虫授粉。昆虫的造访是罕见的,在造访之间植物会增加花蜜的数量,直到达到某种基因决定的水平$x$。当昆虫造访花朵时,它会消耗花蜜;当花蜜的当前量是$z$时,它的消耗率是$r(z)$。因此,访问期间花蜜量的变化率为$\frac{d z}{d t}=-r(z)$。我们假设$r(z)$是$z$的一个递增函数,因此花蜜随着其水平的下降而更难提取。当它的瞬时消耗率下降到$r(z)=\gamma$时,昆虫就会离开,而$\gamma$则取决于整个环境中花蜜的可用性。然后昆虫会转移到另一株植物上,再次寻找花蜜。离开前一朵花后找到另一朵花的平均时间是$\tau$ .


我们假设昆虫选择它们的离开率$\gamma$是为了使它们获得花蜜的平均速率最大化。如果昆虫访问的时间是$T$,植物就会获得好处$B(T)$。这是一个递增但减速的访问时间函数。他们还需要支付与花蜜消耗量相等的每次访问费用——花蜜消耗量等于植物必须生产的额外花蜜量,以将现有的花蜜量补充到原来的水平。植物的收益等于收益减去成本。
为了分析这个模型,假设常驻策略是植物提供花朵到关卡$x$。让$z_\gamma$满足$r\left(z_\gamma\right)=\gamma$,这样$z_\gamma$就是昆虫离开花朵时留下的花蜜量。然后昆虫从每朵花上获得$x-z_\gamma$花蜜。让$T(x, z)$作为花从最初的花蜜级别$x$消耗到级别$z$的时间。这种昆虫消耗花蜜的平均速率是$g(x, \gamma)=\frac{x-z_\gamma}{T\left(x, z_\gamma\right)+\tau}$。这个平均速率可以通过设置$\gamma=\gamma^(x)$来最大化,其中$\gamma^(x)$是方程$g\left(x, \gamma^(x)\right)=\gamma^(x)$的唯一解(McNamara, 1985)。因此,当昆虫的瞬时消耗率下降到$\gamma^(x)$时,花朵中剩下的花蜜量为$z_{\gamma^(x)}$。


现在考虑一种罕见的突变植物 $x^{\prime}$ 当常驻供应级别为 $x$。昆虫会停留一段时间 $T\left(x^{\prime}, z_{\gamma^{+}(x)}\right)$ 在变异的花上。因此,突变体的收益为
$$
W\left(x^{\prime}, x\right)=B\left(T\left(x^{\prime}, z_{\gamma^(x)}\right)\right)-\left(x^{\prime}-z_{\gamma^(x)}\right) .
$$
图 $7.9$ 阐述了寄生策略对一种罕见突变植物平均采蜜率和最佳反应的影响。作为常驻配置级别 $x$ 增加一个昆虫在每朵花上遇到更多的花蜜,它到达一朵花的瞬时消耗速度增加。

经济代写|博弈论代写博弈论代考| choosy,分类,和合作

.


我们现在考虑的市场只有一个物种,群体成员既是服务的提供者又是服务的接受者。具体地说,我们假设个人结对执行一项任务,该任务对结对成员有潜在的好处,但对公共利益的贡献需要个人付出代价(参见3.1节)。假设配对成员的贡献水平存在差异。我们概述了市场环境中变化的存在如何导致对成员贡献的正相关。然后我们认为,这种关联的建立选择了对公共利益的更大贡献;例如,pair成员之间更大的合作。


在市场环境中,对伙伴的挑剔可以通过两种主要方式导致对成员贡献水平的正相关。这两种机制都依赖于这样一个事实:当且仅当伴侣双方都决定这样做时,一对伴侣才会形成或保持在一起


在一对伴侣形成之前,个体可能对未来伴侣可能做出的贡献有所了解。例如,这些信息可能是基于之前与他人的行为而获得的声誉的结果。再假设在寻找伴侣的过程中,个体能够拒绝一个个体,而选择一个更合作的选择。那么,高度合作的群体成员往往会被少数潜在伴侣拒绝,因此有能力挑挑拣拣,而不合作的个体则会被很多人拒绝,没有能力挑挑拣拣。因此,高度合作的个体会倾向于与其他高度合作的个体配对,导致一对中成员的合作能力呈正相关。麦克纳马拉和柯林斯(1990)在相互配偶选择的背景下阐述了这种效应。个体可能会中断与不合作的伙伴的互动,而去寻找另一个更合作的伙伴。如果两个成员都高度合作,那么它们就会倾向于待在一起。相反,如果一个高度合作的人有一个不合作的伴侣,他会“离婚”,并寻找一个更好的伴侣。因此,在合作程度上再次存在正相关关系。McNamara等人(1999a)在两性交配和离婚的背景下阐述了这种效应


有人认为,其他过程可能会导致非亲缘关系之间的积极分类(例如,Joshi等人,2017)


正相关的存在对相关的性状施加了选择压力。Box $7.4$给出了一个例子,在这个例子中,无论相关性的来源是什么,相关性的增加都会导致进化稳定时合作的增加。考虑了两种情况。在第一种情况下,对共同利益的贡献的变化可能被认为是由发展噪声引起的。在第二种情况下,没有发育噪声,但突变等过程维持遗传变异。

经济代写|博弈论代写Game Theory代考

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