I have been studying the paper "Analysis and Control of Networked Game Dynamics via A Microscopic Deterministic Approach" by Tan, Wang and Lu.
In the second page it says that for a 2-strategy game the payoff function of each player from a networked game can be calculated by the last equation shown above.
How do we derive this equation?
I would appreciate any help. Thank you.
The payoff $f_i$ for Player $i$ is computed assuming that she plays each neighbor. So the summation includes one term for each nonzero $a_{ik}$. The payoff from Player $k$ to Player $i$ depends on both players' propensity to use Strategy $C$. The probability that they both play $C$, for example, is $x_ix_k$. If you take the matrix product, you get
$$ ax_ix_k + bx_i(1-x_k) + c(1-x_i)x_k + d(1-x_i)(1-x_j) \enspace, $$
which is the expected payoff for Player $i$.