Optimal approximation ratio for arbitrary population sizes

Determine the optimal approximation ratio of universally strategyproof randomized mechanisms for the truthful interval covering problem for every number $n$ of agents.

Background

For rank lotteries, the paper completely characterizes the optimum: for every n≥4n\ge4, the best ratio is 32−12⌊n/2⌋\frac32-\frac{1}{2\lfloor n/2\rfloor}. Beyond rank lotteries, however, it proves only a four-agent lower bound of $9/8.

The unresolved general problem is to characterize the best achievable ratio when the mechanism may be any universally strategyproof randomized mechanism, rather than only a rank lottery.

References

More generally, determining the optimal approximation ratio of universally strategyproof randomized mechanisms for arbitrary $n$ and studying whether stronger guarantees are possible under truthfulness in expectation remain important directions for future work.

— Optimal Rank Lotteries for Truthful Unit-Interval Covering  (2608.24193 - Voudouris, 25 Aug 2026) in Section 6, “Open questions”