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Equation 17 · The Type Chart Is a Payoff Matrix, and the Metagame Only Half-Solves It

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ϕ(x)=∑ixifi(x)\phi(x) = \sum_i x_i f_i(x)

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Inputs and operationssum_i x_i f_i(x)
Result or conditionphi(x)
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This equation states an equality: the expressions on both sides have the same value under the article’s assumptions. Read the equation part by part below; each part has a contextual explanation and a link to its mathematical background.

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ϕ\phi

Symbol phi

phi is part of the quantity the equation computes from the expression on the right.

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xx

Symbol x

x is an argument of the function-like quantity on the left; its role is set by that function’s stated inputs.

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ii

Symbol i

i appears in the bound of this sum. The bound states where the repeated operation starts, ends, or which values it includes.

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xix_i

Symbol x_i

xix_i is an input to the expression that computes the quantity on the left.

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fif_i

Symbol f_i

the population’s average payoff.

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=

=

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

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subscript

subscript

The lower label selects a particular version, component, or indexed member of the quantity. For example, x₀ and xₜ can be values at different positions.

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ii

Starting index or lower bound: i

This label says where the repeated addition, multiplication, or accumulation starts. Read its value or condition together with the article’s description of the index.

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How to interpret it

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What the article says around this equation

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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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 actual ladder at nearly a quarter of all teams. At the fixed point, every strategy still carrying positive weight earns the same expected payoff against the mix — Great Tusk, Dragonite, and Kingambit are each pinned near a fitness of zero relative to one another — while every extinguished strategy earns strictly less, which is exactly the signature a correctly computed evolutionarily stable mix should have. The computation is not broken. Its answer is just not close to what the ladder shows.

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