Hunting numbers: Optimal strategies in mental searches

Levy flights are a special class of random walk whose step lengths
follow a power-law tailed distribution. There are plenty of empirical
evidences that movement patterns are compatible with Levy flights in
many different contexts where efficiency matters. In particular, Levy
flights appear recurrently in the description of the motion of animals
in real space. Animals explore the environment mainly for searching food
resources, and it is therefore plausible to ascribe the optimality of
their search strategies to a selective evolutionary process. There is
still a lack of knowledge about the statistical properties of search
processes performed by humans, and it is not clear weather humans adopt
search strategies similar to those of animals or not. We report, for the
first time in the literature, empirical evidence that Levy flights
describe searches performed, in mental abstractions of the physical
space, by humans in online games. We analyze the bid histories of online
auctions and show that players, when looking for unknown winning bid
values, follow trajectories in the bid space that can be statistically
described as Levy flights. The power-law scaling can be measured over
several order of magnitude. Differently from previous studies regarding
biological and mobility systems where two orders of magnitude of scaling
can represent a luxury, the power-law decay in bid space explorations
can be clearly observed even over four orders of magnitude. The reason
is that the space is not strictly physical and movements of arbitrary
length can be performed at the same price: players explore the bid space
in a effectively super-diffusive fashion, and steps are made at infinite
velocity. Moreover, we provide a game theoretical model for the
exploration of prices in the auctions.