Theory Of Bar Games

Distributed systems that span multiple administra- tive domains require protocols that tolerate both Byzantine and selfish nodes. This paper offers a theory that can be used to analyze such protocols. The theory systematically extends traditional game theory solution concepts through an ex ante analy- sis that incorporates a rational player's awareness of the possible presence of Byzantine players in the player's utility function. We illustrate our ap- proach by modeling synchronous Terminating Reli- able Broadcast as a game. We show that Dolev and Strong's Byzantine TRB protocol with message au- thentication is not a Nash equilibrium and that ra- tional deviations from it may lead to violation of the TRB safety properties. We present a new TRB protocol with the same asymptotic complexity of Dolev-Strong and prove it to be a Nash equilibrium. Finally, we prove that (k-t) robustness, a recently proposed solution concept for games with Byzantine and rational players, cannot yield an equilibrium in games, such as our TRB game, that model systems where any node may crash and communication is necessary and incurs cost.