Fair leader election for rational agents in asynchronous rings and networks

We study a game theoretic model where a coalition of processors might collude to bias the outcome of the protocol, where we assume that the processors always prefer any legitimate outcome over a non-legitimate one. We show that the problems of Fair Leader Election and Fair Coin Toss are equivalent, and focus on Fair Leader Election. Our main focus is on a directed asynchronous ring of n processors, where we investigate the protocol proposed by Abraham et al. [4] and studied in Afek et al. [5]. We show that in general the protocol is resilient only to sub-linear size coalitions. Specifically, we show that Ω(n log n) randomly located processors or Ω(³ n) adversarially located processors can force any outcome. We complement this by showing that the protocol is resilient to any adversarial coalition of size O(⁴ n). We propose a modification to the protocol, and show that it is resilient to every coalition of size Θ(n), by exhibiting both an attack and a resilience result. For every k ≥ 1, we define a family of graphs G_k that can be simulated by trees where each node in the tree simulates at most k processors. We show that for every graph in G_k, there is no fair leader election protocol that is resilient to coalitions of size k. Our result generalizes a previous result of Abraham et al. [4] that states that for every graph, there is no fair leader election protocol which is resilient to coalitions of size ⌈ⁿ ₂ ⌉.