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Equation 18 · Part 7 · The Type Chart Is a Payoff Matrix, and the Metagame Only Half-Solves It

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DKL(p ∥ q)=∑ipilog⁡2(pi/qi)D_{\mathrm{KL}}(p \,\|\, q) = \sum_i p_i \log_2(p_i/q_i)
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What this part means

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

Its job in the formula

The equals sign connects the complete expression on the left with the complete expression on the right. Both sides must have compatible units.

The passage around this formula

Scoring the fit with Kullback–Leibler divergence, DKL(p ∥ q)D_{\mathrm{KL}}(p \,\|\, q) = ∑i\sum_i pip_i log⁡2(pi/qi)\log_2(p_i/q_i) , from the observed August distribution to each candidate: the solved game scores 10.15 bits, the uniform null scores 0.045 bits, and July’s raw persistence scores 0.00068 bits. On a chi-square test against the eight species’ actual August battle counts (summing to 2,137,239 team-appearances, treating each candidate distribution as the expected shares), the solved game scores approximately 2.3×\times10^{13} — a figure that large only because the model assigns four real, frequently-played species an expected share of essentially zero, and squaring a large real count against a near-zero…

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Learn the underlying idea

An equals sign says that the expression on its left and the expression on its right have the same value under the stated definitions and assumptions.

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Sources cited in the article section

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