经济代写|博弈论代写Game Theory代考|ECON3050

经济代写|博弈论代写Game Theory代考|Co-evolution of Traits

The stability analyses in Chapter 4 deal with the evolution of a single trait, regarding other traits as fixed. That approach is perfectly reasonable when other traits do not interact with the focal trait, but is limiting and can miss important effects when there is a strong interaction. For example, in Section $6.6$ we consider two traits: the degree of prosociality in a group and the dispersal rate between groups. It is perfectly possible to analyse the evolution of each of these traits in isolation, keeping the other trait fixed. However, the kin relatedness in groups depends on the dispersal rate and the degree of prosociality depends on the kin relatedness, so there is an interaction between the traits. As a consequence, when both are allowed to co-evolve there can be two distinct ESSs. $\Lambda$ t onc ESS, bchaviour within a group is coopcrativc and dispersal is low, and at the other behaviour is more selfish and dispersal is high. This result would not be so obvious if the traits were treated in isolation. In this chapter we consider how traits co-evolve when there is interaction between them.

Multidimensional stability is more complicated than stability in one dimension. Even verifying that a singular point is an ESS requires care since a singular point can be a maximum when approached from one direction but a minimum when approached from another. Both the relative speeds with which traits evolve and the genetic covariance between them can affect whether an ESS is convergence stable. We outline some of these issues in Section 6.1.

In two-player interactions the players can be in well-defined roles, with one player in role 1 and the other role 2 (Section 6.2). Furthermore, it may be reasonable to assume that each recognizes its own role. For example, when animals are territorial and an intruder challenges an established territory owner, it will often be reasonable to assume that the intruder is aware that the animal that it is challenging is the territory owner. When there are two clear-cut roles we are concerned with the co-evolution of two traits: behaviour in role 1 and behaviour in role 2 . There is clearly an interaction between these traits: the best action as an intruder depends on the behaviour of the territory owner. When the traits are continuous the slope of the best response in one role to the trait value in the other role can be thought of as a measure of the strength of interaction of the two traits. These best response slopes are also important in determining the convergence stability of a trait combination. We cnd Scction $6.2$ by showing that the interaction based on roles can result in evolutionary predictions that are different from the analogous scenario without roles, using owner-intruder interactions ás an example.

经济代写|博弈论代写Game Theory代考|Role Asymmetries

There are often well-defined roles in two-player interactions, with one player in each role and each player recognizing its own role. For example, in biparental care the roles are male and female. In a dispute over a territory one individual may be the current territory owner and the other an intruder. In such interactions the role of an individual may directly affect that individual’s payoff. For example, possession of a territory may be more valuable to a territory owner than to an intruder if the owner has already invested time and effort in finding out where food sources in the territory are located. However, as the example of territory ownership presented below illustrates, even without payoff asymmetries, the mere existence of well-defined roles can alter evolutionary predictions.

Each individual might carry genes specifying the behaviour in each role, regardless of the current role of that individual. So, for example, females carry genes that specify what to do in the current situation and might also carry genes that specify what they would have done if male (e.g. if the genes are autosomal). Motivated by this, we define a strategy by a pair of traits $\left(x_1, x_2\right)$ to be adopted in each role. Consider a rare mutant strategy $\left(x_1^{\prime}, x_2^{\prime}\right)$ in a population with resident strategy $\left(x_1, x_2\right)$. Since the mutant is rare, a mutant in role 1 is partnered by a resident in role 2 . Let $W_1\left(x_1^{\prime}, x_2\right)$ be the mutant’s local payoff in this role. Similarly $W_2\left(x_2^{\prime}, x_1\right)$ is the mutant’s local payoff in role 2. At any time there are an equal number of individuals in each role, so in the simplest setting where roles are independent of the strategy, half of the mutants are in each role. If this is the case we can define a fitness proxy $W$ as
$$W\left(\left(x_1^{\prime}, x_2^{\prime}\right),\left(x_1, x_2\right)\right)=W_1\left(x_1^{\prime}, x_2\right)+W_2\left(x_2^{\prime}, x_1\right),$$
which we assume is a strong fitness proxy. Let us also assume that the genetic system is such that the evolution of one trait does not constrain the evolution of the other trait. Mutations that alter the value of one of the traits need then not alter the value of the other trait. For this case the best response to a resident strategy should simultaneously maximize the local payoffs $W_1$ and $W_2$. In particular, at a Nash equilibrium $\left(x_1^, x_2^\right)$ we have
$$W_1\left(x_1^{\prime}, x_2^\right) \leq W_1\left(x_1^, x_2^\right) \quad \text { for all } x_1^{\prime},$$ and $$W_2\left(x_2^{\prime}, x_1^\right) \leq W_2\left(x_2^, x_1^\right) \quad \text { for all } x_2^{\prime}$$

经济代写|博弈论代写Game Theory代考|Role Asymmetries

$$W\left(\left(x_1^{\prime}, x_2^{\prime}\right),\left(x_1, x_2\right)\right)=W_1\left(x_1^{\prime}, x_2\right)+W_2\left(x_2^{\prime}, x_1\right),$$

myassignments-help数学代考价格说明

1、客户需提供物理代考的网址，相关账户，以及课程名称，Textbook等相关资料~客服会根据作业数量和持续时间给您定价~使收费透明，让您清楚的知道您的钱花在什么地方。

2、数学代写一般每篇报价约为600—1000rmb，费用根据持续时间、周作业量、成绩要求有所浮动(持续时间越长约便宜、周作业量越多约贵、成绩要求越高越贵)，报价后价格觉得合适，可以先付一周的款，我们帮你试做，满意后再继续，遇到Fail全额退款。

3、myassignments-help公司所有MATH作业代写服务支持付半款，全款，周付款，周付款一方面方便大家查阅自己的分数，一方面也方便大家资金周转，注意:每周固定周一时先预付下周的定金，不付定金不予继续做。物理代写一次性付清打9.5折。

Math作业代写、数学代写常见问题

myassignments-help擅长领域包含但不是全部: