Equation 29 · The Economics and Physical Limits of Running AI Agents at Scale
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 gives an approximation: it relates the quantities while allowing an approximation. 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 E
The expected value operator: the probability-weighted average of the quantity inside its brackets.
Symbol C_n
is part of the quantity the equation computes from the expression on the right.
Symbol i
i appears in the bound of this sum. The bound states where the repeated operation starts, ends, or which values it includes.
Symbol n
n appears in the bound of this sum. The bound states where the repeated operation starts, ends, or which values it includes.
Symbol c_i
occurs above the fraction bar. The numerator is divided by the entire denominator below it.
=
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.
superscript
A raised number can be a power. When it is a label or bound, it selects a case or the upper limit of a sum; the formula’s structure distinguishes these uses.
See an illustrated explanation →Starting index or lower bound: i=1
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.
Ending index or upper bound: n
This label says where the repeated addition, multiplication, or accumulation stops. It sets the last term or end of the range.
Denominator: 1-q_i
The complete quantity below the fraction bar; it must be nonzero for this division.
How to interpret it
With a fixed numerator, increasing a nonzero denominator reduces the fraction. Its accuracy depends on the assumptions and range of use described in the article. Read it with the definitions, units, and assumptions supplied by the article.
What the article says around this equation
If a step fails and is retried with probability , independent of other steps, the expected cost of the trajectory becomes . Holding = q constant across the trajectory — a simplification, since real failure rates are neither constant nor independent — the expected extra cost from retries is , and because increases with i , that extra cost is weighted toward the trajectory’s most expensive, latest steps rather than distributed evenly across it. A one-percent per-step retry rate is a rounding error at step five. Applied at step two hundred of an uncached or lightly cached loop, it is a rounding error on a much larger number.
Sources cited in the article section
These citations give research context. Read each source to check which claims it supports.
Return to The Economics and Physical Limits of Running AI Agents at Scale