Equation 9 · The Type Chart Is a Payoff Matrix, and the Metagame Only Half-Solves It
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Symbol U
U is part of the quantity the equation computes from the expression on the right.
Symbol i
i is an argument of the function-like quantity on the left; its role is set by that function’s stated inputs.
Symbol j
j is an argument of the function-like quantity on the left; its role is set by that function’s stated inputs.
=
The expressions on both sides represent the same quantity under the stated assumptions.
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What the article says around this equation
A matrix of pairwise advantages is an invitation to ask what a population that only cared about that matrix would converge on. Convert E into a genuine zero-sum payoff by taking a log-ratio: U(i,j) = \!E(i,j)/E(j,i) , which is antisymmetric by construction ( U(i,j) = -U(j,i) ) and reads in doublings of relative advantage — U(i,j)=2 means i ’s best attack into j outclasses j ’s best attack into i by a factor of four. This is a standard move in evolutionary game theory, and this piece does not pretend to invent the tool: this house has already used replicator dynamics to model real market selection and a matched null-model discipline to test claims of selection against chance,…
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A matrix of pairwise advantages is an invitation to ask what a population that only cared about that matrix would converge on. Convert E into a genuine zero-sum payoff by taking a log-ratio: U(i,j) = \!E(i,j)/E(j,i) , which is antisymmetric by construction ( U(i,j) = -U(j,i) ) and reads in doublings of relative advantage — U(i,j)=2 means i ’s best attack into j outclasses j ’s best attack into i by a factor of four. This is a standard move in evolutionary game theory, and this piece does not pretend to invent the tool: this house has already used replicator dynamics to model real market selection and a matched null-model discipline to test claims of selection against chance, and both apply directly here rather than needing to be rederived [ 16 , 17 ] .
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