### 经济代写|博弈论代写Game Theory代考|ECON2070

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## 经济代写|博弈论代写Game Theory代考|Replicator Dynamics

It is common in game theory to use a replicator equation as a possible dynamical foundation for the analysis (e.g. Weibull, 1995; Hofbauer and Sigmund, 1998; Broom and Rychtàr, 2013). The equation deals with the dynamics of a fixed number of strategies, or genetic types, in a single large population. Let $n_i(t)$ be the number of individuals using strategy $x_i, i=1, \ldots, k$, and $v_i(t)=n_i(t) / n(t)$ the relative frequency of $x_i$, where $t$ is time measured in generations and $n(t)$ is the total population size. The equation follows from a definition of fitness $w_i(t)$ as the per capita rate of increase of the strategy:
$$w_i(t)=\frac{1}{n_i(t)} \frac{d n_i(t)}{d t} .$$
Note that in general this fitness depends on the composition of the population at time $t$. It is related to but not the same as invasion fitness described in Section 2.1, because it goes beyond the study of the invasion of a rare mutant into a resident population. From the definition we get
$$\frac{d v_i(t)}{d t}=v_i(t)\left(w_i(t)-\bar{w}(t)\right), i=1, \ldots, k,$$
where $\bar{w}=\sum_i v_i w_i$ is the average fitness in the population. Equation (4.2) in Box $4.1$ is the important special case of only two strategies, one of which is present at low frequency.
If the strategies $x_i$ can be treated as real-valued traits, it follows that
$$\frac{d \bar{x}(t)}{d t}=\operatorname{Cov}(w \cdot(t), x \cdot(t)),$$
where $\bar{x}=\sum_i v_i x_i$ is the population mean trait and $\operatorname{Cov}\left(w_{.}, x_{.}\right)=\sum_i v_i\left(w_i-\bar{w}\right)\left(x_i-\right.$ $\bar{x})$ is the population covariance of the trait $x_i$ with fitness $w_i$. This is a version of the celebrated Price equation (Frank, 1995). The derivation of the equations is left as Exercise $4.8$.

The replicator and Price equations have the advantage of a certain generality, in that they follow from a definition of fitness of a strategy as the per-capita rate of increase. They are thus helpful in giving an understanding of how selection operates. However, they do not in themselves solve the difficulties of a population dynamical analysis of a polymorphic population and, in the simple versions given here, they do not deal with issues of population structure and multilocus genotype-phenotype maps. Although opinions vary, one can say that the development of adaptive dynamics has been more strongly oriented towards handling such difficulties.

## 经济代写|博弈论代写Game Theory代考|Games Between Relatives

The study of invasion of rare mutant strategies into a resident population is a leading theme in this chapter. Our approach to games between relatives also develops this idea. For these games, a rare mutant strategy can have an appreciable chance of interacting with other mutant strategies, and need not interact only or predominantly with resident strategies. Relatedness can thus lead to positive assortment of strategies (see below), but games between relatives also include other situations. One example we have encountered is dispersal to reduce kin competition (Section 3.10). Another example is the interaction between parents and offspring about parental investment, where the players of the game are from different generations and can, depending on the precise circumstances, have partly diverging evolutionary interests (see below). The evolutionary analysis of parent-offspring conflicts was initiated by Trivers (1974) and has subsequently been given much attention.

In the original formulation of kin selection by Hamilton (1964), the concept of inclusive fitness was used to study interactions between relatives. In principle the concept has wide applicability (Gardner et al., 2011), but care is sometimes needed for a correct interpretation (Fromhage and Jennions, 2019). However, we will not make use of it here. Let us just note that a main idea of the inclusive fitness approach is to assign fitness effects to an ‘actor’, corresponding to the reproductive consequences of the action for the actor and the actor’s relatives. Alternatively, instead of such an actor-centred approach, one can sum up all reproductive effects for a focal ‘recipient’ individual, and this is referred to as the direct fitness approach (Taylor and Frank, 1996; Taylor et al., 2007). Furthermore, a very straightforward approach that is related to direct fitness is to simply compute invasion fitness of a rare mutant in a resident population. This can be thought of as a ‘gene-centred’ approach and we use it here.

## 经济代写|博弈论代写Game Theory代考|Replicator Dynamics

$$w_i(t)=\frac{1}{n_i(t)} \frac{d n_i(t)}{d t} .$$

$$\frac{d v_i(t)}{d t}=v_i(t)\left(w_i(t)-\bar{w}(t)\right), i=1, \ldots, k,$$

$$\frac{d \bar{x}(t)}{d t}=\operatorname{Cov}(w \cdot(t), x \cdot(t))$$

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