Symbol j

The passage around this formula

Run the continuous replicator equation on the resulting eight-strategy zero-sum game: $\dot{x}_i$ = $x_i$$\big($$f_i(x)$ - $\phi(x)$$\big)$ , where $f_i(x)$ = $\sum_j$ $x_j$\, U(i,j) is species i ’s expected payoff against the current population mix and $\phi(x)$ = $\sum_i$ $x_i$ $f_i(x)$ is the population’s average payoff. Starting from a uniform mix across all eight and integrating to convergence, the system does not settle into anything resembling the real ladder’s spread. It collapses toward a near-monopoly: Great Tusk at 87.71% of the stationary mix, Dragonite at 6.50%, Kingambit at 5.76%, and the other five species asymptoting toward zero, Gholdengo included, despite Gholdengo sitting in second place on the…

Read this part in the article →