Consensus in Equilibrium: Can One Against All Decide Fairly?

Abstract: Is there an equilibrium for distributed consensus when all agents except one collude to steer the decision value towards their preference? If an equilibrium exists, then an $n-1$ size coalition cannot do better by deviating from the algorithm, even if it prefers a different decision value. We show that an equilibrium exists under this condition only if the number of agents in the network is odd and the decision is binary (among two possible input values). That is, in this framework we provide a separation between binary and multi-valued consensus. Moreover, the input and output distribution must be uniform, regardless of the communication model (synchronous or asynchronous). Furthermore, we define a new problem - Resilient Input Sharing (RIS), and use it to find an {\em iff} condition for the $(n-1)$-resilient equilibrium for deterministic binary consensus, essentially showing that an equilibrium for deterministic consensus is equivalent to each agent learning all the other inputs in some strong sense. Finally, we note that $(n-2)$-resilient equilibrium for binary consensus is possible for any $n$. The case of $(n-2)$-resilient equilibrium for \emph{multi-valued} consensus is left open.