Equation 15 · Part 9 · The Type Chart Is a Payoff Matrix, and the Metagame Only Half-Solves It

Starting index or lower bound: j

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

j

What this part means

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.

Its job in the formula

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

Full expressionStarting index or lower bound: jArticle meaning

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 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…

Read this part in the article →

Learn the underlying idea

Σ adds a collection of terms. Π multiplies them. The lower and upper labels tell you which terms belong to the collection.

Open the illustrated sums and products: repeat an operation over an index guide →

See this notation across published equations →

The article lists its research sources here.