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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 经济代写|博弈论代写Game Theory代考|The Evolution of Helping

Helping others can be costly, so that helping behaviour is not expected to evolve unless the cost of providing help is more than compensated for by some benefit. This was illustrated by the Prisoner’s Dilemma game (Section 3.2) where we saw that the strategy of cooperation would not evolve. When pairs are formed from population members at random, defect is the only ESS in that game. There can, however, be reasons to expect a positive assortment where helping individuals are more likely than non-helpers to find themselves interacting with others who help. For example, suppose that offspring have limited dispersal, so that they mature close to their natal environment (population viscosity), and that individuals interact with those around them. Then individuals will be more closely related to those individuals with whom they interact than to average population members. Thus if a mutation arises that provides more help to others than the resident population strategy, then mutants will receive more help from others than residents because some of a mutant individual’s relatives will also have the mutation. Population viscosity is likely to be the most widespread reason for positive assortment in nature, which was originally analysed by Hamilton (1964).

Non-random pairing can thus lead to cooperators doing better than defectors. Our analysis here follows that in McElreath and Boyd (2007). Consider a large population in which each individual either plays cooperate (C) or defect (D) in pair-wise interactions. When pairs are formed there may be a tendency for cooperators to be paired together and for defectors to be paired together. Let $p_{\mathrm{C}}$ be the probability that the partner of a cooperative individual is also cooperative and let $p_{\mathrm{D}}$ be the probability that the partner of a defector is cooperative. Then under positive assortment we would have $p_{\mathrm{C}}>p_{\mathrm{D}}$. In fact the correlation $\rho$ between the level of cooperativeness of pair members can be shown to be exactly $p_C-p_{\mathrm{D}}$ (Exercise 4.10). Conditioning on whether the partner is cooperative or not, the payoff to a cooperator is
$$W(\mathrm{C})=p_C(b-c)+\left(1-p_{\mathrm{C}}\right)(-c) .$$
Similarly, the payoff to a defector is
$$W(\mathrm{D})=p_{\mathrm{D}} b+\left(1-p_{\mathrm{D}}\right) 0 .$$
Thus
$$W(\mathrm{C})>W(\mathrm{D}) \Leftrightarrow\left(p_{\mathrm{C}}-p_{\mathrm{D}}\right) b>c .$$
This shows that if the correlation $\rho=p_{\mathrm{C}}-p_{\mathrm{D}}$ is sufficiently high cooperators can do better than defectors. Box $7.4$ illustrates similar effects of correlation in a model in which traits are continuous.

## 经济代写|博弈论代写Game Theory代考|Hawk–Dove Game Between Relatives

Let us now analyse the Hawk-Dove game, making the same assumptions about relatedness as in Box 4.3. Grafen (1979) originally analysed this game, using the genecentred approach. With the payoffs from Table 3.2, we first examine the invasion of a mixed strategy of playing Hawk with probability $p^{\prime}$ into a resident population using $p$. Instead of eq (3.5) we get
\begin{aligned} W\left(p^{\prime}, p\right)=& W_0+r\left[p^{\prime 2} \frac{1}{2}(V-C)+p^{\prime}\left(1-p^{\prime}\right) V+\left(1-p^{\prime}\right)^2 \frac{1}{2} V\right]+\ &(1-r)\left[p^{\prime} p \frac{1}{2}(V-C)+p^{\prime}(1-p) V+\left(1-p^{\prime}\right)(1-p) \frac{1}{2} V\right] \ =& W_0+\frac{1}{2} V\left[1+(1-r)\left(p^{\prime}-p-\frac{C}{V} p^{\prime} p\right)-r \frac{C}{V} p^{\prime 2}\right] \end{aligned}

Following the same logic as in Box $4.3$, a player using the strategy $p^{\prime}$ has a probhability $r$ of interacting with relatives who also use $p^{\prime}$, and a probability $1-r$ of interacting with random members of the population who use $p$. The fitness gradient (eq (4.4)) is then
$$D(p)=\frac{1}{2} V\left[1-r-(1+r) \frac{C}{V} p\right] .$$
Solving the equation $D\left(p^\right)=0$, we get $$p^=\frac{(1-r) V}{(1+r) C},$$
if $V / C \leq(1+r) /(1-r)$. Compared with the game for unrelated individuals, this probability of playing Hawk is smaller than $V / C$, which can be seen as a tendency towards cooperation. For $V / C>(1+r) /(1-r)$, the equilibrium is $p^=1$. As illustrated in panel (a) of Fig. 4.9, for $r>0$, the equilibrium is a strict maximum of the payoff $W\left(p, p^\right)$, thus satisfying the condition (ES1) for evolutionary stability. Note that this differs from the case $r=0$, for which all strategies are best responses to the ESS.

Even though relatedness has the effect of decreasing the evolutionarily stable intensity of fighting, very dangerous fighting between closely related individuals still occurs in nature. As illustrated in panel (b) of Fig. $4.9$, if two newly emerged honey bee queens meet, they will fight to the death. The explanation lies in the colony life cycle of these insects. When a colony has grown to a certain size, the previous queen leaves the colony with part of the workers, in order to found a new colony (swarming). At this point in time, the colony has reared a number of new queens, but only one of them can inherit the colony ( cf. lethal fighting in Section 3.5). The first new queen to emerge will try to locate the other queen-rearing cells and kill their inhabitants, but if two or more new queens emerge and meet, they fight to the death (Gilley, 2001). Honey bee queens are typically highly related, with $r=0.75$ if they are full sisters (same father), and $r=0.25$ if they are half sisters. The example illustrates that even simple game theory models can deliver surprising and valid predictions.

## 经济代写|博弈论代写Game Theory代考|The Evolution of Helping

$$W(\mathrm{C})=p_C(b-c)+\left(1-p_{\mathrm{C}}\right)(-c) .$$

$$W(\mathrm{D})=p_{\mathrm{D}} b+\left(1-p_{\mathrm{D}}\right) 0$$

$$W(\mathrm{C})>W(\mathrm{D}) \Leftrightarrow\left(p_{\mathrm{C}}-p_{\mathrm{D}}\right) b>c .$$

## 经济代写|博弈论代写Game Theory代考|Hawk–Dove Game Between Relatives

$$W\left(p^{\prime}, p\right)=W_0+r\left[p^{\prime 2} \frac{1}{2}(V-C)+p^{\prime}\left(1-p^{\prime}\right) V+\left(1-p^{\prime}\right)^2 \frac{1}{2} V\right]+\quad(1-r)\left[p^{\prime} p \frac{1}{2}(V-C)+p^{\prime}(1-p) V+\left(1-p^{\prime}\right)(1-p) \frac{1}{2} V\right]=$$

$$D(p)=\frac{1}{2} V\left[1-r-(1+r) \frac{C}{V} p\right] .$$

$$p^{=} \frac{(1-r) V}{(1+r) C}$$

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assignmentutor™您的专属作业导师
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