Rational Fair Consensus in the GOSSIP Model

The \emph{rational fair consensus problem} can be informally defined as follows. Consider a network of $n$ (selfish) \emph{rational agents}, each of them initially supporting a \emph{color} chosen from a finite set $\Sigma$. The goal is to design a protocol that leads the network to a stable monochromatic configuration (i.e. a consensus) such that the probability that the winning color is $c$ is equal to the fraction of the agents that initially support $c$, for any $c \in \Sigma$. Furthermore, this fairness property must be guaranteed (with high probability) even in presence of any fixed \emph{coalition} of rational agents that may deviate from the protocol in order to increase the winning probability of their supported colors. A protocol having this property, in presence of coalitions of size at most $t$, is said to be a \emph{whp\,-$t$-strong equilibrium}. We investigate, for the first time, the rational fair consensus problem in the GOSSIP communication model where, at every round, every agent can actively contact at most one neighbor via a \emph{push$/$pull} operation. We provide a randomized GOSSIP protocol that, starting from any initial color configuration of the complete graph, achieves rational fair consensus within $O(\log n)$ rounds using messages of $O(\log^2n)$ size, w.h.p. More in details, we prove that our protocol is a whp\,-$t$-strong equilibrium for any $t = o(n/\log n)$ and, moreover, it tolerates worst-case permanent faults provided that the number of non-faulty agents is $\Omega(n)$. As far as we know, our protocol is the first solution which avoids any all-to-all communication, thus resulting in $o(n^2)$ message complexity.Comment: Accepted at IPDPS'1

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. \cite{abraham2013distributed} and studied in Afek et al. \cite{afek2014distributed}. We show that in general the protocol is resilient only to sub-linear size coalitions. Specifically, we show that $\Omega(\sqrt{n\log n})$ randomly located processors or $\Omega(\sqrt[3]{n})$ adversarially located processors can force any outcome. We complement this by showing that the protocol is resilient to any adversarial coalition of size $O(\sqrt[4]{n})$. We propose a modification to the protocol, and show that it is resilient to every coalition of size $\Theta(\sqrt{n})$, by exhibiting both an attack and a resilience result. For every $k \geq 1$, we define a family of graphs ${\mathcal{G}}_{k}$ that can be simulated by trees where each node in the tree simulates at most $k$ processors. We show that for every graph in ${\mathcal{G}}_{k}$, 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. \cite{abraham2013distributed} that states that for every graph, there is no fair leader election protocol which is resilient to coalitions of size $\lceil \frac{n}{2} \rceil$.Comment: 48 pages, PODC 201