Simplicity Theory
|
Simplicity, Complexity, Unexpectedness, Cognition,
Probability, Information
by Jean-Louis Dessalles
(created
2008.12.31)
(updated 2010.02.18)
Example: Remarkable
lottery drawings
Simple structures seem unexpected
Though all lottery
combinations have equal probability, individuals judge
simple combinations to be much less
likely (Savoie & Ladouceur
1995; Dessalles, 2006). According to simplicity
theory, this is due to the fact that lottery draws that appear simple are
judged unexpected.
By definition, unexpectedness is the difference
between generation complexity and description complexity: Cw – C. Let’s compute both
terms.
Generation complexity Cw
The generation
complexity of lottery drawings is always about 6 x log2 49, if six numbers are
drawn among 49 (in fact a little less, as a number cannot been
drawn twice).
Description complexity C
Description complexity of lottery
draws depends on their structure. The following table shows description
complexity estimates for various combinations, based on an application of
Leyton’s generative
theory of shape (see Dessalles, 2006 for details).
|
Combinations |
Complexity |
Probability |
|
1 2
3 4 5 6 |
3 |
p0/8´106 |
|
34 35 36 37 38
39 |
6 |
p0/106 |
|
10 11 12 44 45
46 |
11 |
p0/32768 |
|
7 8
9 37 38 39 |
12 |
p0/16384 |
|
8 9
26 27 28 29 |
12 |
p0/16384 |
|
10 20 30 31 32
33 |
12 |
p0/16384 |
|
1 2
5 6 15 49 |
14 |
p0/4096 |
|
. . . |
. . . |
|
|
14 24 36 38 42
44 |
26 |
p0 |
The prediction from formula
p=2–U is that individuals will consider
combinations that are less complex than usual too improbable to be worth the
risk. Indeed, tested subjects considered draws like 1–2–3–4–5–6 as strictly
"impossible" (Dessalles 2006). Most subjects preferred maximally complex
combinations when choosing within a closed list those who would be played for
them.
The description complexity
depends on the structure the subject is able to detect in the combination.
Similarly, when a child sitting in a car sees
66666 on the clock, she can’t help drawing the driver’s attention to the event.
Generating such a number requires five
instantiations (complexity = 5 x log2 10) whereas its description requires only one instantiation
and one iterated copy (complexity = log2 10 + Ccopy).
If we neglect the complexity of copying, formula p=2–U
gives a subjective probability of 10–4, which explains the child’s
behaviour.
Note that the definition of probability p=2–U conflicts with traditional
definitions to be found in complexity theory (Solomonoff 1997),
for which simple objects are more likely. Solomonoff’s
definition, however, pays attention only to generation
complexity and not to description complexity. This is why it makes wrong
predictions in the case of Lottery draws.
Bibliography
Dessalles, J-L. (2006). A structural model of intuitive probability. In D. Fum, F. Del Missier & A. Stocco (Eds.), Proceedings of the seventh International Conference on Cognitive Modeling, 86-91. Trieste, IT: Edizioni Goliardiche.
Savoie, D. & Ladouceur, R. (1995). Évaluation et modification de
conceptions erronées au sujet des loteries. Canadian Journal of Behavioural Science, 27 (2), 199-213.
Solomonoff,
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