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Equation 15 · Part 10 · The Hardest Unsolved Problems in AI Agent Evaluation and Reliability

Starting index or lower bound: i=1

R=1−∑i=1nwi⋅1[trial i failed],R = 1 - \sum_{i=1}^{n} w_i \cdot \mathbb{1}[\text{trial } i \text{ failed}],
i=1i=1

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

i=1 appears in the bound of this sum. The bound states where the repeated operation starts, ends, or which values it includes.

The passage around this formula

and contrast it with a severity-weighted version, R=1−∑i=1nwi⋅1[trial i failed]R = 1 - \sum_{i=1}^{n} w_i \cdot \mathbb{1}[\text{trial } i \text{ failed}]. where wiw_i scales each failure by how costly it actually was. Two agents can share an identical p^\hat{p} of, say, ninety-five percent, while one of them fails harmlessly - an unhelpful but reversible answer - and the other fails catastrophically - an irreversible transaction, a deleted repository, a wrong medical dosage recommendation - on that same five percent. No standard agent benchmark publishes R , because assigning a defensible wiw_i requires a judgment about real-world consequence that a replayable, sandboxed task suite is not built to carry, and because the tasks that would carry the highest weights are, not…

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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.

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