In a fault-tolerant distributed system, different nonfaulty processes may arrive at different values for a given system parameter. To resolve this disagreement, processes must exchange and vote upon their respective local values. Faulty processes may attempt to inhibit agreement by acting in a malicious or "Byzantine" manner. Approximate Agreement defines one form of agreement in which the voted values obtained by the nonfaulty processes need not be identical. Instead, they need only agree to within a predefined tolerance. Approximate Agreement can be achieved by a sequence of convergent voting rounds, in which the range of values held by nonfaulty processes is reduced in each round.Historically, each new convergent voting algorithm has been accompanied by &-hoc proofs of its convergence rate and faulttolerance, using an overly conservative fault model in which aU faults exhibit worst-case Byzantine behavior. This paper presents a general method to quickly determine convergence rate and fault-tolerance for any member of a broad family of convergent voting algorithms. This method is developed under a realistic mixed-mode fault model comprised of asymmetric, symmetric, and benign fault modes. These results are employed to more accurately analyze the properties of several existing voting algorithms, to derive a sub-family of optimal mixed-mode voting algorithms, and to quickly determine the properties of proposed new voting algorithms.

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