SimplicityTheory

Simplicity, Complexity, Unexpectedness, Cognition, Probability, Information

by Jean-Louis Dessalles
(created 31 December 2008)
(updated July 2015)


Coincidences

Amazing coincidences involve events that are simpler when described together.

            

The famous parallel between Abraham Lincoln and John F. Kennedy’s fates leaves no one indifferent, even if one knows that the coincidence should not be regarded as unlikely (Kern & Brow, 2001).

These coincidences should not be considered an objective marvel (Kern & Brown, 2001). However, the fact that it holds a fascination for most of us reveals that some systematic cognitive phenomenon is at work, and this demands an explanation. Let’s list a few factors that are crucial in the Lincoln-Kennedy example.
  1. Analogy: Each common feature between Lincoln and Kennedy’s lives adds to the coincidence. The repeated 100-year interval in the two-president coincidence is an analogy within the analogy.
  2. Association: Non-analogue features, such as the fact that Kennedy was shot in a car named Lincoln, still add to the value of the coincidence.
  3. Prominence: Kennedy and Lincoln are among the foremost US presidents. The coincidence would be less interesting if it involved obscure US presidents, or prominent Uruguayan presidents (for non-Uruguayan citizens).
  4. Round numbers: The Lincoln-Kennedy coincidence would be less interesting if it involved an 87-year interval.
All these parameters have a systematic influence on the perception of coincidences. This is an indication that some genuine cognitive competence is at work and that the surprise generated by coincidences is not a mere side-effect of some general sensitivity to the presence of statistical exceptions. Probabilistic or statistical accounts would fail to account for the above properties.

Simplicity Theory explains the influence of all these factors (Dessalles 2008a).

By definition, unexpectedness U is the difference between generation complexity and description complexity: Cw – C. Let’s compute both terms.

Generation complexity Cw

Let’s call s1 and s2 the two terms of the coincidence. If we make the crucial hypothesis that the two events are independent, the "W-machine" has to do the work twice to generate the two events together, s1 & s1:

Cw(s1 & s2) = Cw(s1) + Cw(s2)

Description complexity C

The "O-machine" requires the following complexity:

C(s1 & s2) < C(s1) + C(s2|s1)

According to ST, the unexpectedness U(s1 & s2) of the double event is the difference Cw(s1 & s2) – C(s1 & s2):

U(s1 & s2) > Cw(s1) + Cw(s2) – C(s1) – C(s2|s1)        (1)

If both events are of comparable complexity: C(s1) ≈ C(s2) and are not unexpected separately: Cw(s1) ≈ C(s1), we get:

U(s1 & s2) > C(s1) – C(s2|s1)        (2)

This expression predicts all aspects of coincidences.
  1. Analogy: It correctly predicts the importance of a close analogy between the two events, as best analogies make C(s1 & s2) minimum (Cornuéjols, 1996). Each new common element, such as the birth date of the successors in the Lincoln-Kennedy parallel, has to be generated twice by the W-machine, but only once by the O-machine. Its impact on the conditional complexity C(s2|s1) is zero. Common elements thus add to the unexpectedness of the situation. An additional prediction is that more complex common elements will add to unexpectedness (e.g. if both presidents had been besides their secretary of defence’s spouse when they were shot).
  2. Association: Expression (2) above predicts that any element of s1 that the O-machine can reuse to generate s2 will add to the surprise. Hence the mention that Kennedy was killed in car named ‘Lincoln’. The W-machine must generate the make of the car and requires several bits which add to Cw(s2) to distinguish it from other makes; this generation is easier for the O-machine when s1 is given as input, as the name ‘Lincoln’ is available for free.
  3. Prominence: Why is it important that Abraham Lincoln and John F. Kennedy be famous? Expression (1) above provides an answer. These two persons exist and are unique in the subject’s world; the W-machine has thus no work to do to generate them. For the O-machine, however, their minimal description may go through determining their social role, US president, and then find them in a ranked list of US-presidents. Hence the importance for the coincidence to involve two prominent figures, as their lower complexity, which amounts to the logarithm of their rank in a list ordered by celebrity, diminishes both C(s1) and C(s2|s1) in expression (1) above, and thus adds to unexpectedness. Conversely, a social role such as president of Uruguay, if more complex for the observer, would make the coincidence less surprising, as it would significantly add to C(s1) and to C(s2|s1).
  4. Round numbers: The presence of the negative term –C(s2|s1) in the expression of unexpectedness explains why the Lincoln-Kennedy story is more interesting as it is, rather than if the repeated time interval had been of 87 years instead of 100. The point is that the cognitive complexity of the program that transforms 1846 into 1946 or 1860 into 1960 is simpler than the program that would transform 1846 into 1933 and 1860 into 1947 (the first program affects only one digit). Alternatively, one can observe that 100, as a number, is simpler than 87. The complexity of an integer n is not larger than log2 n. In the case of round numbers, it is significantly smaller. One million may be concisely defined as 106, or as 1 followed by six copies of 0.
The above considerations generalize to any coincidence. Interestingly, proximity effects apply to coincidences, not only for egocentric distances, but also for allocentric distances (i.e. distances between s1 and s2 in time, space, social networks, ...) as C(s2|s1) crucially depend on them.

Bibliography

Cornuéjols, A. (1996). Analogie, principe d’économie et complexité algorithmique. Actes des 11èmes Journées Françaises de l’Apprentissage. Sète: .

Dessalles, J.-L. (2007). Spontaneous assessment of complexity in the selection of events. Technical Report ParisTech-ENST 2007D011.

Dessalles, J.-L. (2008). Coincidences and the encounter problem: A formal account. In B. C. Love, K. McRae & V. M. Sloutsky (Eds.), Proceedings of the 30th Annual Conference of the Cognitive Science Society, 2134-2139. Austin, TX: Cognitive Science Society.

Dessalles, J.-L. (2008). La pertinence et ses origines cognitives - Nouvelles théories. Paris: Hermes-Science Publications.

Kern, K. & Brown, K. (2001). Using the list of creepy coincidences as an educational opportunity. The history teacher, 34 (4), 531-536.

Warren, C. N. (1934). Modern news reporting. New York: Harper & Brothers, ed. 1959.

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