  We analyze the conditions under which the emergence of frequently observed
echelon formation can be explained solely by the maximization of energy
savings. We consider a two-dimensional multi-agent echelon formation, where
each agent receives a benefit that depends on its position relative to the
others, and adjusts its position to increase this benefit. We analyze the
selfish case where each agent maximizes its own benefit, leading to a
Nash-equilibrium problem, and the collaborative case in which agents maximize
the global benefit of the group. We provide conditions on the benefit function
under which the frequently observed echelon formations cannot be Nash
equilbriums or group optimums.
  We then show that these conditions are satisfied by the conventionally used
fixed-wing wake benefit model. This implies that energy saving alone is not
sufficient to explain the emergence of the migratory formations observed, based
on the fixed-wing model. Hence, either non-aerodynamic aspects or a more
accurate model of bird dynamics should be considered to construct such
formations.
