# 经济代写|博弈论代写Game Theory代考|ECOS3012

## 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.

# 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

$$W\left(x^{\prime}, x\right)=B\left(T\left(x^{\prime}, z_{\gamma^(x)}\right)\right)-\left(x^{\prime}-z_{\gamma^(x)}\right) .$$

## 经济代写|博弈论代写博弈论代考| choosy，分类，和合作

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