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A central result in Fullerton and McAfee’s (1999) analysis of fixed-prize research tournaments shows that if firms’ heterogeneous marginal effort costs are publicly known and the procurer can charge non-discriminatory entry fees, restricting entry to the two most efficient firms is optimal under a (fairly restrictive) sufficient condition on the form of heterogeneity. This note provides a complementary result. I prove a sharp, worst-case bound (across all linear cost structures) for the ratio between the cost of procuring a given total effort from the optimal number of contestants and the corresponding cost for a tournament featuring only the two most efficient firms. The analysis confirms the attractiveness of the smallest possible tournament, with some notable exceptions.