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Related: Discrete vs Continuous Replicator Dynamics

I am trying to perform a derivation of the discrete time replicator dynamics, but I am unable to get through Cressman's derivation in "Evolutionary Dynamics and Extensive form games" on page 21

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Basic notations:


$S = \{e_1, \ldots, e_n\}$ are pure strategies

$\pi(e_i, e_j)$ payoff of player using strategy $e_i$ versus strategy $e_j$

$p \in \Delta^n = \{(p_1, \ldots, p_n) | p_i \geq 0, \sum_i p_i = 1\}$ mixed strategy

$\pi(p, \hat p) = p \cdot A\hat p = \sum\limits_{i,j = 1}^n p_i a_{ij} \hat p_j$ is the payoff of strategy $p$ versus $\hat p$


Derivation of the discrete replicator dynamic:


Let $n_i$ be the number of individuals using $e_i$ at generation $t$.

Then $n_i' = n_i (e_i \cdot Ap)$ be the expected number of individuals at generation $t+1$

Let $p_i = \dfrac{n_i}{\sum_j n_j}$ and $p'_i = n_ie_i \cdot \dfrac{Ap}{\sum\limits_j n_j e_j} \cdot Ap = (\sum_j n_j)p_ie_i \cdot \dfrac{Ap}{\sum\limits_j n_j}\sum_j p_je_j \cdot Ap = p_i(e_i \cdot \dfrac{Ap}{p}\cdot Ap)$

Then the standard discrete time replicator dynamic is $p'_i = p_i \dfrac{e_i \cdot Ap}{p\cdot Ap}$


Questions: There are numerous moving parts in this derivation and little explanation as to where each item came about


  1. Just to make sure, $n_i' = n_i(t+1)$ right?

  2. Why is that $n_i' = n_i (e_i \cdot Ap)$ is the expected number of individuals at the next generation? This is just the number of individuals at the current generation times the payoff to pure strategy $e_i$, is there a derivation for $n_i'$?

  3. I understand $p_i$ is the fraction of players using $i$ versus all the other strategies. But can someone explain how $p_i'$ came about, where are all its pieces? I am especially confused about the part where it says $Ap/p$, because $p$ is a vector and you can't divide it...

  4. Is there another derivation of the discrete time replicator dynamic?

Any help is appreciated.

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  • $\begingroup$ I think you misunderstood some of the nominators/denominators in the text you cited. In the cited text, the whole stuff after the / sign is the denominator of the fraction. So there is no division by p, but division by (p \cdot Ap), which is a scalar. $\endgroup$
    – Andreas
    Feb 13 '18 at 14:57

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