Optimal Decision Strategies in Byzantine Environments

A Boolean value of given a priori probability distribution is transmitted to a deciding agent by several processes. Each process fails independently with given probability, and faulty processes behave in a Byzantine way. A deciding agent has to make a decision concerning the transmitted value on the basis of messages obtained by processes. We construct a deterministic decision strategy which has the provably highest probability of correctness. It computes the decision in time linear in the number of processes. Decision optimality may be alternatively approached from a local, rather than global, point of view. Instead of maximizing the total probability of correctness of a decision strategy, we may try to find, for every set of values conveyed by processes, the conditionally most probable original value that could yield this set. We call such a strategy locally optimal, as it locally optimizes the probability of a decision, given a set of relayed values, disregarding the impact of such a choice on the overall probability of correctness. We construct a locally optimal decision strategy which again computes the decision value in time linear in the number of processes. We establish the surprising fact that, in general, local probability maximization may lead to a decision strategy which does not have the highest probability of correctness. However, if the probability distribution of the Boolean value to be conveyed is uniform, and all processes have the same failure probability smaller than 1 2 , this anomaly does not occur. We first design and analyze our strategies in the synchronous setting and then show how they should be modified to work in asynchronous