Classical spatial models of two-party competition typically predict convergence to the median voter, yet real-world party systems often exhibit persistent and asymmetric polarization. We develop a spatial model of two-party competition in which voters evaluate parties through general satisfaction functions, and a width parameter $q$ captures how tolerant they are of ideological distance. This parameter governs the balance between centripetal and centrifugal incentives and acts as the bifurcation parameter governing equilibrium configurations. Under mild regularity assumptions, we characterize Nash equilibria through center-distance coordinates, which separate the endogenous political center from polarization. When the voter density is symmetric, the reduced equilibrium condition exhibits a generic supercritical pitchfork bifurcation at a critical value $q_{c}$. Above $q_{c}$, the unique stable equilibrium features convergence to the center, recovering the classical median voter result, whereas below it two symmetric polarized equilibria arise. Asymmetry in the voter distribution unfolds the pitchfork, producing drift in the endogenous center and asymmetric polarized equilibria. The resulting equilibrium diagram has an S-shaped geometry that generates hysteresis, allowing polarization to persist even after tolerance returns to levels that would support convergence in a symmetric environment. In the high-tolerance regime, we show that the unique non-polarized equilibrium converges to the mean of the voter distribution, while the median is recovered only under symmetry. Hence, unlike the Hotelling--Downs model, where convergence to the median is universal, the median voter appears here as an asymptotic benchmark rather than a robust predictor.

翻译：经典的两党竞争空间模型通常预测政党会向中间选民趋同，然而现实中的政党体系常常表现出持续且不对称的极化现象。我们构建了一个两党竞争的空间模型，其中选民通过广义满意度函数评估政党，宽度参数 $q$ 刻画了他们对意识形态距离的容忍程度。该参数调控着向心激励与离心激励之间的平衡，并作为支配均衡构型的分岔参数。在温和的正则性假设下，我们通过中心-距离坐标刻画纳什均衡，该坐标将内生的政治中心与极化程度分离开来。当选民密度分布对称时，简化后的均衡条件在临界值 $q_{c}$ 处呈现通用的超临界叉形分岔。在 $q_{c}$ 之上，唯一的稳定均衡表现为向中心趋同，重现了经典的中间选民结果；而在 $q_{c}$ 之下，则会产生两个对称的极化均衡。选民分布的非对称性会展开该叉形结构，导致内生中心发生漂移并产生不对称的极化均衡。由此得到的均衡图具有S形几何结构，从而产生滞后效应——即使容忍度恢复到在对称环境下足以支持趋同的水平，极化仍可能持续存在。在高容忍度区域，我们证明唯一的非极化均衡会收敛于选民分布的均值，而中间选民结果仅在对称条件下得以恢复。因此，与霍特林-唐斯模型中向中间选民的趋同具有普适性不同，本模型中中间选民更多地表现为一个渐近基准，而非稳健的预测结果。