How the Republic of Venice chose its Doge: lot-based elections and supermajority rule

We study a family of voting rules inspired by the peculiar protocol used for over 500 years by the Republic of Venice to elect its Doge. Lot-based indirect elections have two main features: a pool of delegates is chosen by lot out of a general assembly, and then they vote in a single winner election with qualified majority. Under the assumption that the assembly is divided into two factions, we characterise the win probability of the minority and show that these features promote a more equitable allocation of political representation, striking a balance between protecting the minority and giving proper recognition to the majority. We then consider this family of voting procedures from a constitutional perspective: we analyse how the electoral result varies with the college size and the winning threshold in order to understand how these two parameters can be tuned when drawing up electoral law. We find that minorities are better off with larger majority thresholds. The role of the college size, on the other hand, is ambiguous: a smaller college size offers more protection to sparse minorities; for more sizeable ones, it depends instead on the qualified majority required for the election.