Whist tournaments for \( v \) players, \( \mathrm{Wh}(v) \), are known to exist for all \( v \equiv 0, 1 \pmod{4} \). In this paper, a new specialization of whist tournament design, namely a \({balanced\; whist\; tournament}\), is introduced. We establish that balanced whist tournaments on \( v \) players, \( \mathrm{BWh}(v) \), exist for several infinite classes of \( v \). An adaptation of a classic construction due to R. C. Bose and J. M. Cameron enables us to establish that \( \mathrm{BWh}(4n + 1) \) exist whenever \( 4n + 1 \) is a prime or a prime power. It is also established that \( \mathrm{BWh}(4n) \) exist for \( 4n = 2^k \), with \( k \equiv 0 \pmod{2, 3 \text{ or } 5} \). We demonstrate that a \( \mathrm{BWh}(4n + 1) \) is equivalent to a conference matrix of order \( 4n + 2 \). Consequently, a necessary condition for the existence of a \( \mathrm{BWh}(4n + 1) \) is that \( 4n + 1 \) is a product of primes each of which is \( \equiv 1 \pmod{4} \). Thus, in particular, \( \mathrm{BWh}(21) \) and \( \mathrm{BWh}(33) \) do not exist. Specific examples of \( \mathrm{BWh}(v) \) are given for \( v = 4, 8, 9, 20, 24, 32 \). It is also shown that a \( \mathrm{BWh}(12) \) does not exist.