On the (Im)possibility of Game-Theoretically Fair Leader Election Protocols

We consider the problem of electing a leader among n parties with the guarantee that each (honest) party has a reasonable probability of being elected, even in the presence of a coalition that controls a subset of parties, trying to bias the output. This notion is called “game-theoretic fairness” because such protocols ensure that following the honest behavior is an equilibrium and also the best response for every party and coalition. In the two-party case, Blum’s commit-and-reveal protocol (where if one party aborts, then the other is declared the leader) satisfies this notion and it is also known that one-way functions are necessary. Recent works study this problem in the multi-party setting. They show that composing Blum’s 2-party protocol for logn rounds in a tournament-tree-style manner results with perfect game-theoretic fairness: each honest party has probability ⩾1/n of being elected as leader, no matter how large the coalition is. Logarithmic round complexity is also shown to be necessary if we require perfect fairness against a coalition of size n-1. Relaxing the above two requirements, i.e., settling for approximate game-theoretic fairness and guaranteeing fairness against only constant fraction size coalitions, it is known that there are O(log^∗n) round protocols. This leaves many open problems, in particular, whether one can go below logarithmic round complexity by relaxing only one of the strong requirements from above. We manage to resolve this problem for commit-and-reveal style protocols, showing thatΩ(logn/loglogn) rounds are necessary if we settle for approximate fairness against very large (more than constant fraction) coalitions;Ω(logn) rounds are necessary if we settle for perfect fairness against n^ε size coalitions (for any constant ε>0). Ω(logn/loglogn) rounds are necessary if we settle for approximate fairness against very large (more than constant fraction) coalitions; Ω(logn) rounds are necessary if we settle for perfect fairness against n^ε size coalitions (for any constant ε>0). These show that both relaxations made in prior works are necessary to go below logarithmic round complexity. Lastly, we provide several additional upper and lower bounds for the case of single-round commit-and-reveal style protocols.