Characterization and uniqueness of equilibria in locational choice problems

This paper considers the problem of characterizing equilibria in markets involving n producers and a differentiated product. Each producer chooses a position in a set Ω taking into account the distribution of consumer ideals. Subsequently, an equilibrium is reached if no producer has incentive to unilaterally reposition. In this work, we consider the two extreme cases of this problem: In the first case, producers may be ignorant about consumer ideals and play conservatively by assuming a uniform distribution. We show that the rich class of equilibria that exists for this case can be narrowed down to a unique equilibrium when inequity considerations are taken into account. First, this involves showing that a so-called "minimal inequity equilibrium" is obtained via solution of a specially constructed nonlinear program. Subsequently, we display a unique closed form solution for this nonlinear program, which provides considerable insight that cannot be obtained by usual numerical methods. On the other extreme, when complete knowledge of consumer preferences can be assumed, we consider an atomic distribution of ideals. Under such a distribution, we generate a new large class of equilibria.