We study strategyproof mechanisms for locating a single facility on a circle so as to serve a set of strategic agents under the utilitarian social cost objective. We analyze a simple parity-dependent mechanism that randomizes between the Random Dictator (RD) mechanism and the Proportional Circle Distance (PCD) mechanism. For an odd number \(n\) of agents, this mechanism mixes RD and PCD with equal probability, as originally proposed by Rogowski and Dziubi{ń}ski (IJCAI 2025). For even \(n\), it mixes RD with a random-deletion extension of PCD, in which one agent is removed uniformly at random before PCD is applied to the remaining agents. Our main result shows that this mechanism is strategyproof and achieves an approximation ratio of \(\frac32\) for every \(n\ge 3\). This improves the \(\frac74\) upper bound of Rogowski and Dziubi{ń}ski, which applied only to odd \(n\), and extends the guarantee to even numbers of agents. Finally, we establish a lower bound of \(\frac{11}{10}\) on the approximation ratio of any randomized strategyproof mechanism, improving on the previous lower bound of \(1.0456\) due to Meir (SAGT 2019).

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