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Equation 10 · Part 5 · Comparing Frontier Model Pricing Without Comparing Apples to Oranges

Symbol gamma_i

Ci  =  κi[(1−hi) piin+hi piin γi]  +  ρi piout.C_i \;=\; \kappa_i\Big[(1-h_i)\,p^{\text{in}}_i + h_i\, p^{\text{in}}_i\,\gamma_i\Big] \;+\; \rho_i\, p^{\text{out}}_i .
γi\gamma_i

What this part means

the cache-read multiplier.

Its job in the formula

gammaia_i is one of the signed contributions combined to compute the quantity on the left.

Where the article explains it

For vendor i , let piinp^{\text{in}}_i and pioutp^{\text{out}}_i be the published input and output rates; let κi\kappa_i be a tokenizer expansion factor, the number of tokens vendor i ’s own tokenizer needs to encode one fixed reference passage, normalised so κi\kappa_i = 1 for whichever vendor is used as the baseline; let hih_i be the fraction of input served from cache and γi\gamma_i the cache-read multiplier; and let ρi\rho_i be the ratio of total output tokens generated, visible answer plus hidden reasoning, to the visible answer alone.

The passage around this formula

…the number of tokens vendor i ’s own tokenizer needs to encode one fixed reference passage, normalised so κi\kappa_i = 1 for whichever vendor is used as the baseline; let hih_i be the fraction of input served from cache and γi\gamma_i the cache-read multiplier; and let ρi\rho_i be the ratio of total output tokens generated, visible answer plus hidden reasoning, to the visible answer alone. A workload’s realised cost per completed task is then approximately Ci  =  κi[(1−hi) piin+hi piin γi]  +  ρi pioutC_i \;=\; \kappa_i\Big[(1-h_i)\,p^{\text{in}}_i + h_i\, p^{\text{in}}_i\,\gamma_i\Big] \;+\; \rho_i\, p^{\text{out}}_i . Reading piinp^{\text{in}}_i and pioutp^{\text{out}}_i…

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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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Sources cited in the article section

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