Equation 7 · The Type Chart Is a Payoff Matrix, and the Metagame Only Half-Solves It
What does this equation mean?
Read the formula alongside the article passage below. Each part has a deeper page with its role in the equation, the supporting passage and nearby citations.
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.
Read it piece by piece
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.
Symbol E
E is an input to the expression that computes the quantity on the left.
=
The expressions on both sides represent the same quantity under the stated assumptions.
See an illustrated explanation →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.
How to interpret it
Read it with the definitions, units, and assumptions supplied by the article.
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,…
Read the full surrounding passage
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 ] .
For background, read the article’s source list.
Return to The Type Chart Is a Payoff Matrix, and the Metagame Only Half-Solves It