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Equation 15 · Part 7 · The Type Chart Is a Payoff Matrix, and the Metagame Only Half-Solves It

=

fi(x)=∑jxj U(i,j)f_i(x) = \sum_j x_j\, U(i,j)
=

What this part means

The expressions on both sides represent the same quantity under the stated assumptions.

Its job in the formula

The equals sign connects the complete expression on the left with the complete expression on the right. Both sides must have compatible units.

The passage around this formula

Run the continuous replicator equation on the resulting eight-strategy zero-sum game: x˙i\dot{x}_i = xix_i(\big(fi(x)f_i(x) - ϕ(x)\phi(x))\big) , where fi(x)f_i(x) = ∑j\sum_j xjx_j\, U(i,j) is species i ’s expected payoff against the current population mix and ϕ(x)\phi(x) = ∑i\sum_i xix_i fi(x)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…

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Learn the underlying idea

An equals sign says that the expression on its left and the expression on its right have the same value under the stated definitions and assumptions.

Open the illustrated equality: what the equals sign claims guide →

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