Conceptualizing voter choice for directional and discounting models of two-candidate spatial competition in terms of shadow candidates

In contrast to the traditional modeling of voter choice based on proximity, under directional models, selection of candidates is based on the direction and/or intensity of change from a status quo or neutral point. Voter choice can also be modeled as representing both approaches, e.g., as a directional model with proximity restraint, or alternatively, in terms of proximity to discounted positions. We provide a unified perspective for these seemingly disparate models in terms of what we call “shadow” positions. We demonstrate that voter choice in a variety of spatial models including directional components can be viewed as proximity-based choices. Voters choose the candidate whose shadow is nearer, where shadow locations are defined by a simple transformation. We apply this approach to equilibrium analysis, showing that results for a discounted proximity model can be carried over – via shadows – to a variety of directional models.