Constructing certainty by using binary search boundaries in the unknown

When I face something truly unknown, or help someone else make a decision, I reshape the situation so it becomes compatible with a binary search. I set extreme seed boundaries, which give me something broad to compare against, and narrow from there until the answer is tight and scoped.

The original dialogue. The asking side never asks for the time again after the first question; it only proposes times to compare against.

The person answering is not missing the answer. They are missing a format they can answer in.

The core claim

Producing a value is hard: there is no reference to hold it against. Judging a value is easy: the reference is handed over. The method converts one generative question into a chain of comparative ones, and each answer is nearly free.

The dialogue as a search

Three moves do the work:

Seed bounds (23:30, 10:00) are absurd on purpose. Their answers are near-certain, so they cost nothing and fence in an unbounded space. Computer science calls this step unbounded search: first find a bracket, then bisect inside it.¹
Interior probes (17:00, 13:00) are not midpoints. They land where the asker suspects the edge is, which is faster than halving. When answers are noisy, the efficient version of bisection² probes the median of what you currently believe, not the middle of the interval.³
The commit takes the middle of what is left. 15:00 is a decision, not a discovery.

Why it works

Every “Before X?” asks for one value of a curve: the chance the event has happened by X. The person can’t recite the curve, but they can read any single point of it on request.

This is why the method beats a direct question:

Comparison is cheap; magnitude is expensive. People judge a stimulus relative to a reference far more reliably than they identify its absolute value, a limit known since the 1950s,⁴ and explained by a model in which the units of judgement are differences, not magnitudes.⁵
Graded answers carry more than a bit. “Of course” and “likely” are different heights on the curve, so each answer is worth more than a yes or no. Psychophysics has used the same trick for decades to find perceptual thresholds, placing each next trial where the last answers leave the most doubt.⁶
The person holds a distribution, not a number. Asked the same question twice, people give two different answers, and their average beats either one, as if each answer were a sample from an internal distribution.⁷
The chart exposes a gap. Taking the middle of 13:00 to 17:00 gives 15:00, but the answers put the median nearer 15:30. The leaks below return to this.

Where the idea already lives

The closest match is decision analysis, which built this exact protocol for eliciting expert forecasts at the Stanford Research Institute in the 1970s: iterated “more or less than X?” questions, with the extremes encoded first.⁸ Each of the other fields owns one piece of it.

Picked greedily, each for what it adds beyond the ones before it, five sources are enough:

Spetzler and Staël von Holstein: the full method and its debiasing.⁸
Stewart, Brown and Chater: the mechanism, why comparisons are easy.⁵
Tversky and Kahneman: the main failure mode, anchoring.⁹
Probabilistic bisection: the formal version with noisy, graded answers.³
Plato’s Meno: the interpersonal layer, the person discovering they knew.¹⁰

The rest repeat these. Hubbard popularized the elicitation protocol,¹¹ and the mathematics of searching when some answers lie was worked out as Ulam’s game.¹²

What is mine: the literature runs this as expert elicitation by an analyst. My version is an everyday conversational tool, and its main payoff, showing someone they knew more than they thought, is the least studied part.

Where it leaks

The method has four known leaks. Each has a one-line fix.

The protocol

Six steps for any conversation where someone says “I don’t know.”

Never ask for the value. Ask only “more or less than X?”
Seed two absurd bounds, one on each side. The answers should be obvious.
Name the bracket. “So you have an idea.” The person learns they were never ignorant.¹⁶
Probe where you suspect the edge, alternating sides to blunt anchoring.
Stop when the window is small enough to act on, or when answers turn to “not sure”: you have reached the soft middle of the curve.
Commit at the 50% point and report the window.

In one line: each question reads one point of $F(x) = P(T \le x)$ , the chance of arriving by $x$ , and the commitment is $F^{-1}(0.5)$ .

References

Bentley, J. L., & Yao, A. C.-C. (1976). An almost optimal algorithm for unbounded searching. Information Processing Letters, 5(3), 82–87. doi:10.1016/0020-0190(76)90071-5. ↩
Horstein, M. (1963). Sequential transmission using noiseless feedback. IEEE Transactions on Information Theory, 9(3), 136–143. doi:10.1109/TIT.1963.1057832. ↩
Waeber, R., Frazier, P. I., & Henderson, S. G. (2013). Bisection search with noisy responses. SIAM Journal on Control and Optimization, 51(3), 2261–2279. doi:10.1137/120861898. ↩ ↩²
Miller, G. A. (1956). The magical number seven, plus or minus two: Some limits on our capacity for processing information. , 63(2), 81–97. .

Anchoring	“Before 17:00?” pulls the estimate toward 17:00. Asking whether a value is higher or lower than X, then asking for an estimate,¹³ is exactly the classic anchoring setup,⁹ and the effect replicates across dozens of labs¹⁴	Seed with extremes first, alternate sides, let the person propose the next value
Collapse	A 4-hour window is reported as one point, and the uncertainty vanishes	Report both: “15:00, window 13:00 to 17:00”
Midpoint is not median	“Impossible” and “likely” are different heights, so the middle of the window sits below the 50% point	Close with one last probe: “More or less likely than not by 15:00?”
Constructed answer	The commitment partly creates the certainty it reports¹⁵	For a decision this is the goal. For a forecast, say it was built, not found