Simplicity Theory

Simplicity, Complexity, Unexpectedness, Cognition, Probability, Information

by Jean-Louis Dessalles
(created 2008.12.31)

(updated 2010.02.17)

Example: The ‘You? Here?’ effect

The encounter problem is spectacular, as it provides the best evidence that the human mind is sensitive to description complexity.

Interest grows with the complexity of the place and with the simplicity of the encountered person.

Fortuitous encounters are all the more unexpected that the place is complex and the encountered person is simple.

Example: In 2008, I was travelling in Uganda, I joined a group of tourists to visit the Muchison Falls National park. I discovered that a German couple in the group were the best friends of my nephew’s girlfriend. The coincidence made quite an impression on us.

Fortuitous encounters seem to be an exception to the rule of closeness, as the interest grows this time with the remoteness of the place! A proper application of the notion of unexpectedness, however, restores the prediction.

Interest grows with the complexity of the place and with the simplicity of the encountered person. The complexity of the place l is the relevant factor, not the distance: a big distant airport may be less complex than the backyard of an obscure building of a lost suburb a few kilometres away. The simplicity of the encountered person P is the relevant factor, not her closeness. Running into a celebrity may be as unexpected as running into a close colleague. These phenomena are correctly predicted by the fact that unexpectedness varies, as we will show, as:

U = C(l) – C(P)

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

Calculus 1:

Let us compute the unexpectedness of the sequence ego*P*l*l(ego)*l(P). Here, l(ego) and l(P) designate the presence of self and of the encountered person P at location l. We may write:

C_w(ego*P*l*l(ego)*l(P)) = C_w(ego) + C_w(P) + C_w(l) + C_w(l(ego)|l) + C_w(l(P)|l&l(ego)&P)

Note: Irrelevant elements in conditional complexity expressions are omitted for the sake of clarity. For instance, C_w(P|ego) = C_w(P).

The complexity of ego is supposed to be zero. For the W-machine, C_w(P) = 0 and C_w(l) = 0, as P and l are supposed to exist: the "world-machine" has no work to do to produce them. In the ‘world’, P’s and ego’s common presence at l are supposed to be independent: C_w(l(P)|l&l(ego)&P) = C_w(l(P)|l&P) (note that this is a crucial assumption). If ego and P play symmetrical roles, the W-machine requires C_w(ego*P*l*l(ego)*l(P)) = 2 C_w(l(ego)|l) to generate the encounter situation.

One way to assess the term C_w(l(ego)|l) is to consider the complexity of the decisions that ego had to take to end up in location l. If l is not itself unexpected, then in most cases, C_w(l(ego)|l) = C(l) and amounts to the minimum size of a set of directions to reach l (exception: if l is materially difficult to reach). Finally:

C_w(ego*P*l*l(ego)*l(P)) = 2 C(l)

The O-machine demands complexity C(P) to determine P and C(l) to determine l. On the other hand, C(l(ego)|l) < C(l|l) + C(ego|l) = 0. C(l(P)|l(ego)&P) = 0 as well, as P is directly observed by ego in l. We eventually get:

C(ego*P*l*l(ego)*l(P)) = C(P) + C(l)

and thus as announced:

U(ego*P*l*l(ego)*l(P)) = C(l) – C(P)

Calculus 2:

An alternative computation of C_w goes over the sequence ego*l*l(ego)*l(P)*P. This time, P is first determined by her position l(P) = l(ego). Now C_w(l(P)|l(ego)) is zero, but C_w(P|l(P)) is not. To instantiate P, the W-machine has to distinguish among all individuals that may happen to be in l. One procedure to do so consists in checking local people first, by delimiting an area of radius R around l and considering all people living within it. Then R is increased until the actual P is reached. In this computation, P’s home and l play symmetrical roles, so that C_w(P|l(P)) = C(l) + c (l is supposed to be as complex for P to reach as for ego). The constant c depends on the spatial density of people. The resulting unexpectedness C(l) + c – C(P) is similar to what we previously obtained.

Note that this second computation still holds when the encounter occurs in the vicinity of ego’s home. In this case, C_w(l(ego)) and C(l(ego)) are negligible, but C_w(l(P)) is close to C(h(P)) (the complexity of P’s home) and the unexpectedness amounts to C(h(P)) + c – C(P). It is indeed quite a coincidence to meet a celebrity in front of one’s home.

Bibliography

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.

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