This paper, authored by Martin Osborne and Al Slivinski, develops and analyzes a model of electoral competition where citizens can choose to run as candidates. The model considers both plurality rule and majority rule with runoffs, examining the conditions under which different numbers of candidates can coexist in equilibrium. Key findings include: 1. **Plurality Rule**: - A one-candidate equilibrium exists if the benefit of winning ($b$) is small relative to the cost of entry ($c$). - A two-candidate equilibrium exists if $b \geq 2(c - e_p(F))$, where $e_p(F)$ is the critical value for a citizen to enter and win outright. - Equilibria with more than two candidates are possible, but the number of candidates depends positively on the ratio of $b$ to $c$. 2. **Runoff System**: - A two-candidate equilibrium exists if $2(c - e_r(F)) \leq b \leq 4c$, where $e_r(F)$ is the supremum of values for which a citizen with ideal position in $(m - \epsilon, m + \epsilon)$ can win outright. - Equilibria with more than two candidates are more likely under a runoff system, but the conditions for such equilibria are more complex. 3. **Strategic Behavior**: - Citizens may enter the competition to affect the outcome, even if they have no chance of winning. - The incentives for entering differ under plurality rule and runoff systems, with the former discouraging splintering votes among extreme candidates. 4. **Conclusion**: - The model provides insights into the diversity of political competitions, supporting hypotheses like Duverger’s Law and the role of strategic calculations in election outcomes. The paper also discusses the implications of entry costs, the distribution of citizens' preferences, and the impact of uncertainty on equilibrium outcomes.This paper, authored by Martin Osborne and Al Slivinski, develops and analyzes a model of electoral competition where citizens can choose to run as candidates. The model considers both plurality rule and majority rule with runoffs, examining the conditions under which different numbers of candidates can coexist in equilibrium. Key findings include: 1. **Plurality Rule**: - A one-candidate equilibrium exists if the benefit of winning ($b$) is small relative to the cost of entry ($c$). - A two-candidate equilibrium exists if $b \geq 2(c - e_p(F))$, where $e_p(F)$ is the critical value for a citizen to enter and win outright. - Equilibria with more than two candidates are possible, but the number of candidates depends positively on the ratio of $b$ to $c$. 2. **Runoff System**: - A two-candidate equilibrium exists if $2(c - e_r(F)) \leq b \leq 4c$, where $e_r(F)$ is the supremum of values for which a citizen with ideal position in $(m - \epsilon, m + \epsilon)$ can win outright. - Equilibria with more than two candidates are more likely under a runoff system, but the conditions for such equilibria are more complex. 3. **Strategic Behavior**: - Citizens may enter the competition to affect the outcome, even if they have no chance of winning. - The incentives for entering differ under plurality rule and runoff systems, with the former discouraging splintering votes among extreme candidates. 4. **Conclusion**: - The model provides insights into the diversity of political competitions, supporting hypotheses like Duverger’s Law and the role of strategic calculations in election outcomes. The paper also discusses the implications of entry costs, the distribution of citizens' preferences, and the impact of uncertainty on equilibrium outcomes.