  The Levy walk in which the frequency of occurrence of step lengths follows a
power-law distribution, can be observed in the migratory behavior of organisms
at various levels. Levy walks with power exponents close to 2 are observed, and
the reasons are unclear. This study aims to propose a model that universally
generates inverse square Levy walks (called Cauchy walks) and to identify the
conditions under which Cauchy walks appear. We demonstrate that Cauchy walks
emerge universally in goal-oriented tasks. We use the term "goal-oriented" when
the goal is clear, but this can be achieved in different ways, which cannot be
uniquely determined. We performed a simulation in which an agent observed the
data generated from a probability distribution in a two-dimensional space and
successively estimated the central coordinates of that probability
distribution. The agent has a model of probability distribution as a hypothesis
for data-generating distribution and can modify the model such that each time a
data point is observed, thereby increasing the estimated probability of
occurrence of the observed data. To achieve this, the center coordinates of the
model must be moved closer to those of the observed data. However, in the case
of a two-dimensional space, arbitrariness arises in the direction of correction
of the center; this task is goal oriented. We analyze two cases: a strategy
that allocates the amount of modification randomly in the x- and y-directions,
and a strategy that determines allocation such that movement is minimized. The
results reveal that when a random strategy is used, the Cauchy walk appears.
When the minimum strategy is used, the Brownian walk appears. The presence or
absence of the constraint of minimizing the amount of movement may be a factor
that causes the difference between Brownian and Levy walks.
