We put forward a new approach for the design of efficient multiparty protocols: 1. Design a protocol π for a small number of parties (say, 3 or 4) which achieves security against a single corrupted party. Such protocols are typically easy to construct, as they may employ techniques that do not scale well with the number of corrupted parties. 2. Recursively compose π with itself to obtain an efficient n-party protocol which achieves security against a constant fraction of corrupted parties. The second step of our approach combines the "player emulation" technique of Hirt and Maurer (J. Cryptology, 2000) with constructions of logarithmic-depth formulae which compute threshold functions using only constant fan-in threshold gates. Using this approach, we simplify and improve on previous results in cryptography and distributed computing. In particular: - We provide conceptually simple constructions of efficient protocols for Secure Multiparty Computation (MPC) in the presence of an honest majority, as well as broadcast protocols from point-to-point channels and a 2-cast primitive. - We obtain new results on MPC over blackbox groups and other algebraic structures. The above results rely on the following complexity-theoretic contributions, which may be of independent interest: - We show that for every j,k ∈ ℕ such that m Delta equal to k-1/j-1 is an integer, there is an explicit (poly(n)-time) construction of a logarithmic-depth formula which computes a good approximation of an (n/m)-out-of-n threshold function using only j-out-of-k threshold gates and no constants. For the special case of n-bit majority from 3-bit majority gates, a non-explicit construction follows from the work of Valiant (J. Algorithms, 1984). - For this special case, we provide an explicit construction with a better approximation than for the general threshold case, and also an exact explicit construction based on standard complexity-theoretic or cryptographic assumptions.