Competition between choices over time

Introduction Paper (PDF)

Measuring choices over time implies examining competition between alternates.

This is a fairly obvious statement. However, some of the mathematical properties of this system are less well known. These inform the expected behaviour of observations, helping us correctly specify null hypotheses.

  • The proportion of {shall, will} utterances where shall is chosen, p(shall | {shall, will}), is in competition with the alternative probability of will (they are mutually exclusive) and bounded on a probabilistic scale.
  • The probability associated with each member of a set of alternates X = {xi}, which we might write as p(xi | X), is bounded, 0 ≤ p(xi | X) ≤ 1, and exhaustive, Σp(xi | X) = 1.

A bounded system behaves differently from an unbounded one. Every child knows that a ball bouncing in an alley behaves differently than in an open playground. ‘Walls’ direct motion toward the centre.

In this short paper we discuss two properties of competitive choice:

  1. the tendency for change to be S-shaped rather than linear, and
  2. how this has an impact on confidence intervals.


S curves and Wilson intervals

We can sketch the overall behaviour of the system by plotting Wilson intervals for the ‘S’ curve (below). We have plotted a logistic curve for p (k = 1) and added intervals for n = 10 and n = 100. Observe that with a small n the confidence interval is large and more highly skewed. The difference (w⁺ – w⁻) is greatest for p = 0.5.

Logistic curve (k = 1) with Wilson score intervals for n = 10, 100.


  1. The S curve
  2. Poisson error bars
  3. S curves and Wilson intervals


Wallis, S.A. 2010. Competition between choices over time. London: Survey of English Usage, UCL.

See also

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