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Consider a population structured in groups of two individuals. Individuals' interactions follow an additive prisoner's dilemma:

\begin{array}{c c c} & C & D \\ \text{Cooperate (} C \text{)} & b -c & -c \\ \text{Defect (} D\text{)} & b & 0 \end{array} where $b$ is the benefit and $c$ is the cost. The payoffs are to the player on the left. I need to calculate the regression of individual fitness on individual phenotype, $\beta (w_i, p_i)$, where $p_i = 0$ if $i$ defects and $p_i = 1$ if $i$ cooperates. Note that I need to calculate the regression within groups rather than between groups.

I thought that the right way of calculating $\beta(w_i,p_i)$ would be to vary the strategy of the focal individual while keeping the strategy of the other individual constant (my thought was that the other individual becomes the environment for the focal individual). Because switching to cooperation causes an individual to lose $-c$ in fitness (if we keep the strategy of the other individual constant), I thought that $\beta(w_i,p_i) = -c$. But, to my surprise, $\beta(w_i,p_i) = -b -c$. To show why, McElreath & Boyd ("Mathematical Models of Social Evolution", p. 242) draw the following graph:

Let $V(X|Y)$ be the payoff of individual $X$ when it interacts with $Y$ (in other words, $X$ is the focal individual; $Y$ is the other individual). I understand that McElreath and Boyd calculated the regression coefficient by computing $V(C|D) - V(D|C)$ --- as opposed to $V(C|C) - V(D|C)$, where you keep the strategy of the other individual constant. My question is why this is the right way of calculating $\beta(w_i,p_i)$.

Consider a population structured in groups of two individuals. Individuals' interactions follow an additive prisoner's dilemma:

\begin{array}{c |c |c|} & C & D \\ \hline \text{Cooperate (} C \text{)} & b -c & -c \\ \hline \text{Defect (} D\text{)} & b & 0 \\ \hline \end{array} where $b$ is the benefit and $c$ is the cost. The payoffs are to the player on the left. I need to calculate the regression of individual fitness on individual phenotype, $\beta (w_i, p_i)$, where $p_i = 0$ if $i$ defects and $p_i = 1$ if $i$ cooperates. Note that I need to calculate the regression within groups rather than between groups.

I thought that the right way of calculating $\beta(w_i,p_i)$ would be to vary the strategy of the focal individual while keeping the strategy of the other individual constant (my thought was that the other individual becomes the environment for the focal individual). Because switching to cooperation causes an individual to lose $-c$ in fitness (if we keep the strategy of the other individual constant), I thought that $\beta(w_i,p_i) = -c$. But, to my surprise, $\beta(w_i,p_i) = -b -c$. To show why, McElreath & Boyd ("Mathematical Models of Social Evolution", p. 242) draw the following graph:

Let $V(X|Y)$ be the payoff of individual $X$ when it interacts with $Y$ (in other words, $X$ is the focal individual; $Y$ is the other individual). I understand that McElreath and Boyd calculated the regression coefficient by computing $V(C|D) - V(D|C)$ --- as opposed to $V(C|C) - V(D|C)$, where you keep the strategy of the other individual constant. My question is why this is the right way of calculating $\beta(w_i,p_i)$.

Consider a population structured in groups of two individuals. Individuals' interactions follow an additive prisoner's dilemma:

\begin{array}{c c c} & C & D \\ \text{Cooperate (} C \text{)} & b -c & -c \\ \text{Defect (} D\text{)} & b & 0 \end{array} where $b$ is the benefit and $c$ is the cost. The payoffs are to the player on the left. I need to calculate the regression of individual fitness on individual phenotype, $\beta (w_i, p_i)$, where $p_i = 0$ if $i$ defects and $p_i = 1$ if $i$ cooperates. Note that I need to calculate the regression within groups rather than between groups.

I thought that the right way of calculating $\beta(w_i,p_i)$ would be to vary the strategy of the focal individual while keeping the strategy of the other individual constant (my thought was that the other individual becomes the environment for the focal individual). Because switching to cooperation causes an individual to lose $-c$ in fitness (if we keep the strategy of the other individual constant), I thought that $\beta(w_i,p_i) = -c$. But, to my surprise, $\beta(w_i,p_i) = -b -c$. To show why, McElreath & Boyd ("Mathematical Models of Social Evolution", p. 242) draw the following graph:

Let $V(X|Y)$ be the payoff of individual $X$ when it interacts with $Y$ (in other words, $X$ is the focal individual; $Y$ is the other individual). I understand that McElreath and Boyd calculated the regression coefficient by computing $V(C|D) - V(D|C)$ --- as opposed to $V(C|C) - V(D|C)$, where you keep the strategy of the other individual constant. My question is why this is the right way of calculating $\beta(w_i,p_i)$.

Here is another way of formulating my question. $w_i$ can be written in terms of $p_i$ as follows (note that $p_i$ can only assume two values, 0 and 1):

$w_i (0) = w_0 + kb$

$w_i (1) = w_0 -c + (k-1)b$

where $w_0$ is the baseline fitness and $k$ is the number of altruists in the group (so, $k$ can be 0, 1, or 2). If we keep $k$ constant, $\beta(w_i, p_i)$ is equal to $-b -c$ (for $\beta(w_i, p_i) = w_i(1) - w_i(0)$). My question is: why should we keep $k$ constant when calculating the regression coefficient?

Consider a population structured in groups of two individuals. Individuals' interactions follow an additive prisoner's dilemma:

\begin{array}{c c c} & C & D \\ \text{Cooperate (} C \text{)} & b -c & -c \\ \text{Defect (} D\text{)} & b & 0 \end{array} where $b$ is the benefit and $c$ is the cost. The payoffs are to the player on the left. I need to calculate the regression of individual fitness on individual phenotype, $\beta (w_i, p_i)$, where $p_i = 0$ if $i$ defects and $p_i = 1$ if $i$ cooperates. Note that I need to calculate the regression within groups rather than between groups.

I thought that the right way of calculating $\beta(w_i,p_i)$ would be to vary the strategy of the focal individual while keeping the strategy of the other individual constant (my thought was that the other individual becomes the environment for the focal individual). Because switching to cooperation causes an individual to lose $-c$ in fitness (if we keep the strategy of the other individual constant), I thought that $\beta(w_i,p_i) = -c$. But, to my surprise, $\beta(w_i,p_i) = -b -c$. To show why, McElreath & Boyd ("Mathematical Models of Social Evolution", p. 242) draw the following graph:

Let $V(X|Y)$ be the payoff of individual $X$ when it interacts with $Y$ (in other words, $X$ is the focal individual; $Y$ is the other individual). I understand that McElreath and Boyd calculated the regression coefficient by computing $V(C|D) - V(D|C)$ --- as opposed to $V(C|C) - V(D|C)$, where you keep the strategy of the other individual constant. My question is why this is the right way of calculating $\beta(w_i,p_i)$.

Consider a population structured in groups of two individuals. Individuals' interactions follow an additive prisoner's dilemma:

\begin{array}{c c c} & C & D \\ \text{Cooperate (} C \text{)} & b -c & -c \\ \text{Defect (} D\text{)} & b & 0 \end{array} where $b$ is the benefit and $c$ is the cost. The payoffs are to the player on the left. I need to calculate the regression of individual fitness on individual phenotype, $\beta (w_i, p_i)$, where $p_i = 0$ if $i$ defects and $p_i = 1$ if $i$ cooperates. Note that I need to calculate the regression within groups rather than between groups.

I thought that the right way of calculating $\beta(w_i,p_i)$ would be to vary the strategy of the focal individual while keeping the strategy of the other individual constant (my thought was that the other individual becomes the environment for the focal individual). Because switching to cooperation causes an individual to lose $-c$ in fitness (if we keep the strategy of the other individual constant), I thought that $\beta(w_i,p_i) = -c$. But, to my surprise, $\beta(w_i,p_i) = -b -c$. To show why, McElreath & Boyd ("Mathematical Models of Social Evolution", p. 242) draw the following graph:

Let $V(X|Y)$ be the payoff of individual $X$ when it interacts with $Y$ (in other words, $X$ is the focal individual; $Y$ is the other individual). I understand that McElreath and Boyd calculated the regression coefficient by computing $V(C|D) - V(D|C)$ --- as opposed to $V(C|C) - V(D|C)$, where you keep the strategy of the other individual constant. My question is why this is the right way of calculating $\beta(w_i,p_i)$.

Consider a population structured in groups of two individuals. Individuals' interactions follow an additive prisoner's dilemma:

\begin{array}{c c c} & C & D \\ \text{Cooperate (} C \text{)} & b -c & -c \\ \text{Defect (} D\text{)} & b & 0 \end{array} where $b$ is the benefit and $c$ is the cost. The payoffs are to the player on the left. I need to calculate the regression of individual fitness on individual phenotype, $\beta (w_i, p_i)$, where $p_i = 0$ if $i$ defects and $p_i = 1$ if $i$ cooperates. Note that I need to calculate the regression within groups rather than between groups.

I thought that the right way of calculating $\beta(w_i,p_i)$ would be to vary the strategy of the focal individual while keeping the strategy of the other individual constant (my thought was that the other individual becomes the environment for the focal individual). Because switching to cooperation causes an individual to lose $-c$ in fitness (if we keep the strategy of the other individual constant), I thought that $\beta(w_i,p_i) = -c$. But, to my surprise, $\beta(w_i,p_i) = -b -c$. To show why, McElreath & Boyd ("Mathematical Models of Social Evolution", p. 242) draw the following graph:

Let $V(X|Y)$ be the payoff of individual $X$ when it interacts with $Y$ (in other words, $X$ is the focal individual; $Y$ is the other individual). I understand that McElreath and Boyd calculated the regression coefficient by computing $V(C|D) - V(D|C)$ --- as opposed to $V(C|C) - V(D|C)$, where you keep the strategy of the other individual constant. My question is why this is the right way of calculating $\beta(w_i,p_i)$.

Here is another way of formulating my question. $w_i$ can be written in terms of $p_i$ as follows (note that $p_i$ can only assume two values, 0 and 1):

$w_i (0) = w_0 + kb$

$w_i (1) = w_0 -c + (k-1)b$

where $w_0$ is the baseline fitness and $k$ is the number of altruists in the group (so, $k$ can be 0, 1, or 2). If we keep $k$ constant, $\beta(w_i, p_i)$ is equal to $-b -c$ (for $\beta(w_i, p_i) = w_i(1) - w_i(0)$). My question is: why should we keep $k$ constant when calculating the regression coefficient?