Proof-of-Stake systems randomly choose, on each round, one of the participants as a consensus leader that extends the chain with the next block such that the selection probability is proportional to the owned stake. However, distributed random number generation is notoriously difficult. Systems that derive randomness from the previous blocks are completely insecure; solutions that provide secure random selection are inefficient due to their high communication complexity; and approaches that balance security and performance exhibit selection bias. When block creation is rewarded with new stake, even a minor bias can have a severe cumulative effect. In this paper, we propose Robust Round Robin, a new consensus scheme that addresses this selection problem. We create reliable long-term identities by bootstrapping from an existing infrastructure, such as Intel's SGX processors, or by mining them starting from an initial fair distribution. For leader selection we use a deterministic approach. On each round, we select a set of the previously created identities as consensus leader candidates in round robin manner. Because simple round-robin alone is vulnerable to attacks and offers poor liveness, we complement such deterministic selection policy with a lightweight endorsement mechanism that is an interactive protocol between the leader candidates and a small subset of other system participants. Our solution has low good efficiency as it requires no expensive distributed randomness generation and it provides block creation fairness which is crucial in deployments that reward it with new stake.