## 经济代写|博弈论代写Game Theory代考|Convergence Stability with a Role Asymmetry

Suppose the trait in each role is continuous. Let the payoff to a rare mutant with strategy $\left(x_1^{\prime}, x_2^{\prime}\right)$ in a population with resident strategy $\left(x_1, x_2\right)$ be given by eq (6.4). Assume that for each $x_2$ the local payoff $W_1\left(x_1^{\prime}, x_2\right)$ has a strict maximum at $x_1^{\prime}=\hat{b}_1\left(x_2\right)$ satisfying
$$\frac{\partial W_1}{\partial x_1^{\prime}}\left(\hat{b}_1\left(x_2\right), x_2\right)=0 \text { and } \frac{\partial^2 W_1}{\partial x_1^{\prime 2}}\left(\hat{b}_1\left(x_2\right), x_2\right)<0 .$$
The local best response $\hat{b}_2\left(x_1\right)$ in role 2 has analogous properties. Let $x_1^=\hat{b}_1\left(x_2^\right)$ and $x_2^=\hat{b}_2\left(x_1^\right)$, so that $\left(x_1^, x_2^\right)$ is an ESS. We analyse the convergence stability of this equilibrium in terms of the slopes of the local best response functions.

We first give an intuitive argument. Suppose that initially trait 1 has a value $x_1(0)$ that is close to, but not equal to $x_1^$. Trait 2 then evolves to be the local best response to this trait 1 value, after which trait 1 evolves to be the best response to this trait 2 value, and so on. In this way we obtain a sequence $x_1(0), x_1(1), x_1(2), \ldots$ of trait 1 values where $x_1(n)=\hat{b}_1\left(\hat{b}_2\left(x_1(n-1)\right)\right)$. As Fig. $6.1$ illustrates, the sequence of trait 1 values converges on $x_1^$ if the function $B$ given by $\left.B\left(x_1\right)=\hat{b}_1\left(\hat{b}_2\left(x_1\right)\right)\right)$ has a derivative that satisfies $B^{\prime}\left(x_1^\right)<1$. Conversely, if $B^{\prime}\left(x_1^\right)>1$ the sequence moves away from $x_1^$. Since $B^{\prime}\left(x_1^\right)=\hat{b}_1^{\prime}\left(x_2^\right) \hat{b}_2^{\prime}\left(x_1^\right)$ this suggests that
$\hat{b}_1^{\prime}\left(x_2^\right) \hat{b}_2^{\prime}\left(x_1^\right)<1 \Longrightarrow\left(x_1^*, x_2^*\right)$ is convergence stable $\hat{b}_1^{\prime}\left(x_2^*\right) \hat{b}_2^{\prime}\left(x_1^*\right)>1 \Longrightarrow\left(x_1^, x_2^\right)$ is not convergence stable.

## 经济代写|博弈论代写Game Theory代考|Territory Owner Versus Intruder

Suppose that a territory owner and an intruder contest the owner’s territory. Each must choose either to play Hawk or Dove. The outcomes and payoffs to each are exactly as in the standard Hawk-Dove game (Section 3.5). In particular possession of this territory is worth $V$ to both contestants and the cost of losing a fight is $C$. We assume that $V<C$, so that in the game without role asymmetries $p^*=V / C$ is the unique Nash equilibrium and is also an ESS (Section 4.1).

For this game there are two traits. Trait 1 is the probability, $p_1$, of playing Hawk when in the role of territory owner and trait 2 is the probability, $p_2$, of playing Hawk as an intruder. Suppose that the resident strategy is $\left(p_1, p_2\right)$. Then the best trait 1 response (i.e. the best response as an owner) only depends on the resident trait 2 value (behaviour of intruders). Following the analysis of the standard HawkDove game we see that it is given by $\hat{p}_1=1$ if $p_2V / C$, and is any probability of playing Hawk if $p_2=V / C$. The best trait 2 response is similarly $\hat{p}_2=1$ if $p_1V / C$, and is any probability of playing Hawk if $p_1=V / C$. At a Nash equilibrium $p_1^$ must be a trait 1 best response to $p_2^$ and vice versa. It can be seen that there are three Nash equilibria, $(1,0),(0,1)$, and $(V / C, V / C)$. At the first Nash equilibrium owners always play Hawk and intruders always play Dove. Since the action in each role is the unique best response to the resident strategy, this Nash equilibrium is also an ESS. Similarly, the strategy $(0,1)$ that specifies play Dove as owner and Hawk as intruder is an ESS. At the third Nash equilibrium both owners and intruders play Hawk with probability $V / C$. In Section $4.1$ we saw that the strategy $p^*=V / C$ is an ESS for the standard HawkDove game. This was hecause, although mutants do equally well as residents against residents, once a mutation becomes common mutants play against other mutants and do less well than residents do against mutants. In the current game consider a mutation that, say, changes the aggressiveness when in the role of territory owner; i.e. a mutant’s strategy is $\left(p_1^{\prime}, V / C\right)$ where $p_1^{\prime} \neq V / C$. As before, mutants do equally well as residents against residents. Now, however, when two mutants play against one another, one is in the role of the intruder and plays Hawk with probability $V / C$ just as residents do. Thus mutants do as well as residents even when they become common. Mutants can therefore invade by random drift and the Nash equilibrium is not an ESS.

## 经济代写|博弈论代写Game Theory代考|Territory Owner Versus Intruder

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