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    $\begingroup$ To remove the risk of a second No voter randomly picking the initial number, you can instruct that any No voter must pick a higher number than the one they heard. $\endgroup$
    – Ralph J
    Commented Dec 30, 2022 at 5:12
  • $\begingroup$ To remove the risk of the first-and-only No voter learning that he was it, have "voter 2" pick a multiplier at random, which he privately gives to First and Last. They each multiply their numbers by that multiplier, and "voter 3" hears, privately, the product from First and then from Last, and announces "equal" or "not equal". Not knowing the multiplier that V2 picked, he won't know if he's the sole No, and nobody else hears more than one number. $\endgroup$
    – Ralph J
    Commented Dec 30, 2022 at 5:19
  • $\begingroup$ @RalphJ Good suggestions! I think the second one works, but the first one leaks some (probabilistic) information about the number of 'no' voters. But if we choose large numbers then the probability of collision is negligible anyway. $\endgroup$
    – 2012rcampion
    Commented Dec 31, 2022 at 16:57
  • $\begingroup$ Agree - everyone using moderately large numbers each with several factors makes inferences about others' behavior less likely. Meaning that those who don't want their choice to be inferred, have motivation to use numbers /increments/multipliers like 120 - 240, instead of 3-5. $\endgroup$
    – Ralph J
    Commented Dec 31, 2022 at 21:08
  • $\begingroup$ @RalphJ I don't know what you mean about numbers with several factors or increments, isn't the best strategy just to choose a number uniformly at random from the available range? $\endgroup$
    – 2012rcampion
    Commented Dec 31, 2022 at 21:59