Despite common perception, existing theoretical literature lacks a complete formal argument regarding the relationship between the electoral rule disproportionality and platform polarization. In this article, we build a model that incorporates the disproportionality of the electoral system in a standard Downsian electoral competition setup with mainly, but not necessarily purely, policy-motivated parties. We first show that in equilibrium, platform polarization is decreasing in the level of the electoral rule disproportionality. We then argue that the number of parties has a positive effect on platform polarization when polarization is measured by the distance between the two most distant platforms. This effect does not hold when polarization is measured by the widely used Dalton index. Constructing a data set covering more than 300 elections, our main theoretical findings are empirically supported, pointing toward the electoral rule disproportionality as a major determinant of polarization.

Competition between groups often involves prizes that have both a public and a private component. The exact nature of the prize not only affects the strategic choice of the sharing rules determining its allocation but also gives rise to an interesting phenomenon not observed when the prize is either purely public or purely private. Indeed, we show that in the two-groups contest, for most degrees of privateness of the prize, the large group uses its sharing rule as a mean to exclude the small group from the competition, a situation called monopolization. Conversely, there is a degree of relative privateness above which the small group, besides being active, even outperforms the large group in terms of winning probabilities, giving rise to the celebrated group size paradox.

We study parties' optimal ideological cohesion across electoral rules, when the following trade-off is present: A more heterogenous set of candidates is electorally appealing (catch-all party), yet, it serves policy-related goals less efficiently. When the rule becomes more disproportional, thus inducing a more favorable seat allocation for the winner, the first effect is amplified, incentivizing parties to be less cohesive. We provide empirical support using a unique data-set that records candidates' ideological positions in Finnish municipal elections. Exploiting an exogenous change of electoral rule disproportionality at different population thresholds, we identify the causal effect of electoral rules on parties' cohesion.

How does electoral rule disproportionality affect the structure of the party system (i.e. the number and the policy platforms of the competing parties)? By studying a model where both party entry and platform choice are endogenous we are able to provide a unified theory: An increasing electoral rule disproportionality exhibits: a) a first-order negative effect on platform polarization, b) a second-order negative effect on the number of parties (as polarization decreases, centrist parties are squeezed between other contenders and prefer not to enter), and c) an additional third-order negative effect on polarization via the reduction of the number of parties. We then conduct a laboratory experiment and strongly confirm the theoretical predictions of the model.