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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|演化博弈论代写game theory in biology代考|Variation has Consequences

In natural populations there is almost always a considerable amount of betweenindividual variation in behaviour as well as in other traits. For instance, for quantitative traits one finds that the standard deviation is often around $10 \%$ or more of the mean value (Houle, 1992). Furthermore, many estimates of the so-called repeatability of behaviours such as foraging, aggressiveness, and parental behaviour have been collected. The repeatability is the proportion of the variance in behaviour that is due to differences between individuals, and it ranges from around $0.1$ to $0.8$, with an average around $0.4$ (Bell et al., 2009). For game theory this means that, first, we should not assume that variation is small and, second, that it is realistic to examine questions of consistency in behaviour and individual reputation.

One can conceptualize variation in behaviour as resulting from differences in the underlying strategy, in how the strategy is implemented, or from state differences. Strategy variation arises from underlying genetic variation and the implementation of a strategy is affected by developmental plasticity and noise. Concerning states, even if two individuals follow the same strategy, differences in previous experience give rise to state differences, so that the individuals may take different actions in the same current situation. Some phenotypic variation may thus be for adaptive reasons and some not, but regardless of its source, its existence has consequences for game theory.
Most models in the literature ignore that there may be large amounts of genetic variation at evolutionary stability. In considering the evolutionary stability of a Nash equilibrium the effect of variation due to mutation is certainly considered, but in contrast to what is the case in real populations, the amount of variation is assumed to be small; typically the fate of rare mutants is considered (Section 4.1).

In Section $3.11$ we gave examples of state-dependent behaviour when there are state differences. In these examples the corresponding models with no state differences give similar predictions to the models with variation and a decision threshold when these latter models assume little variation in state, although there were some changes. For example, in the Hawk-Dove game variation in fighting ability reduces the frequency of fights and the payoffs of the alternative actions are no longer equalized at evolutionary stability: Hawks do better than Doves. As we will see, in other cases variation can produce even qualitative shifts in perspective.

## 经济代写|演化博弈论代写game theory in biology代考|Variation and the Stability of Equilibria

Consider the trust game illustrated in Fig. 7.2. In this two-player game, one individual is assigned the role of Player 1 and the other Player 2. Player 1 must first decide whether to trust or reject Player 2. If Player 2 is rejected both players receive payoff $s>0$ and the game ends. If Player 2 is trusted this player decides whether to cooperate with Playcr 1 or to defect. If coopcration is chosen cach reccives a payoff of $r$, where $0<s<r<1$. If Player 2 defects Player 1 gets a payoff of 0 and Player 2 a payoff of 1 .

In this game it is clear that Player 2 should defect if trusted since $r<1$. This means that Player 1 will get 0 if Player 2 is trusted and so should reject Player 2 . Thus both players receeive payoff $s$, whereas they could potentially have recéived the larger payoff of $r$. This game can be thus be seen as a version of the Prisoner’s Dilemma game in which choices are made sequentially rather then simultaneously.

Suppose that the resident strategy is for Player 1 to reject Player 2 and for Player 2 to defect if trusted. Then the resident strategy is at a Nash equilibrium. It is not, however, an ESS. To see this consider the mutant strategy that specifies rejection when in the role of Player 1 but cooperation if trusted in the Player 2 role. This strategy has exactly the same payoff as the resident strategy even when common, and so will not be selected against. Formally, although condition (ES2)(i) holds, condition (ES2)(ii) does not. The problem here is that although the mutant and resident strategies differ in their choice of action in the role of Player 2, they never get to carry out this action as they are never trusted. In fact there is no ESS for this game.

Now suppose that in this resident population there is some mechanism that maintains variation in the action of Player 1 individuals, so that occasionally a Player 1 trusts its partner. This variation might be due to the occasional error, to mutation maintaining genetic variation, or any other source. When a Player 2 individual is trusted it gets to carry out its action. If the Player 2 individual is a mutant that cooperates it receives the payoff $r$, while a resident Player 2 defects and obtains the higher payoff of 1 . Thus mutants of this kind are selected against. In other words, the occasional error by Player 1 stabilizes the Nash equilibrium against invasion by mutants.

A strategy is a rule that specifies the action taken for every possible state of the organism. In games in which there is a sequence of choices the state of an organism is usually defined by the sequence of actions that have occurred up until the present time. So, for example, in the trust game, being trusted is a state for Player 2 . If in a population following a Nash equilibrium strategy some state is never reached, then there is no selection pressure on the action taken in that state. This means that mutations that change the action in this state alone are never selected against, so that the Nash equilibrium cannot be an ESS. If the Nash equilibrium is stabilized by occasional errors, as for the trust game, the resident strategy is referred to as a limit ESS. We return to this topic in Chapter 8 when we consider sequences of choices in more detail.

Variation can similarly remove the neutrality from signalling games (Section 7.4). Unless there is some possibility of receiving a particular signal there is no selection pressure on the response to that signal. Variation can ensure that all signals are possible.

# 博弈论代考

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## 有限元方法代写

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## MATLAB代写

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