Equation 15 · Part 1 · The Type Chart Is a Payoff Matrix, and the Metagame Only Half-Solves It

Symbol f_i

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

f_i

What this part means

the population’s average payoff.

Its job in the formula

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

Full expressionSymbol f_iArticle meaning

Where the article explains it

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.

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…

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Learn the underlying idea

A subscript is a label attached below a symbol. It often selects a time step, component, category, or member of a sequence.

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