Before revision:  Besides being of independent theoretical interest, our results enable the network to run an efficient validator election protocol that simultaneously achieves the PJR property and a constant-factor approximation for maximin support, thus offering strong theoretical guarantees on decentralization and security .
After revision:  Besides being of independent theoretical interest, our results enable the network to run an efficient validator election protocol with formal and verifiable guarantees on security and proportionality. We propose a specific protocol that can be successfully implemented in spite of the stringent time constraints of a blockchain architecture, and that will be the basis for an implementation in the Polkadot network, launched in 2020.