Let X be any (pre-specified) single-winner voting system which inputs votes and outputs a winner, a second-place finisher, and a third-place finisher; and let ε>0 be any arbitrarily-small positive real number. Construct a single-winner voting system (which includes randomness) with the following properties.
Please also criticize your solution and expound upon its practical limitations.
This scheme will convert any member X of a wide class of voting systems into a voting system in which voting honestly is the (generally uniquely) best voting strategy, thus essentially totally evading the Gibbard-Satterthwaite impossibility theorem, probably the single most important theorem in voting theory!!
The catch is, each ballot will now have two parts, (i) the "be honest here" part and (ii) the other part. A smart voter will indeed be honest in part (i), but the problem is that the winner is determined by both parts, not just by part (i), of the ballots.
The main idea behind this solution is due to Forest Simmons, 19 Nov 2009. (For a quite different utility-reading method, which seems of less practical value, see puzzle #104.) However we've also incorporated contributions by Jobst Heitzig, Jameson Quinn, and Warren D. Smith.
The rules of our new voting method (which is not voting method X, but is derived from X) are as follows.
We use as a lemma the known fact that in 2-choice simple-majority elections, honest voting is the optimal strategy; and if there is uncertainty about the other votes, it is the uniquely optimal strategy.
This has assumed that voting a sincere range-style ballot is "free" – that it doesn't cost time or "cognitive energy," either for candidate research or for evaluation.
Of course, in a large democracy, voting at all would scarcely ever be selfishly rational if it weren't free, and people still vote. So that assumption can't be too wrong. Still, I think we'd need ε>0.001 to motivate sufficient voters to be sufficiently careful with their range voting.
Also, note that to the extent anybody extracts and publishes useful data from the honest range totals, we unfortunately add strategic incentives to that vote. (We were pretending "strategic voters" were solely interested in who won. Trouble is, they might also be interested in influencing who gets the "moral victory" of winning the unofficial "honest vote.") For instance, if I vote strategically on the X-style sub-ballot, I may choose to vote the same strategy on the "honest" range-style sub-ballot to inflate the published "sincere score" of my candidate, even though this risks sacrificing my chance to favorably-change the real winner.
Finally, note that, presuming we're electing some sort of official who has given term of office, this system could, instead of using randomness, split the term of office between two candidates. A relatively-short deterministic term in office for the "backup" candidate elected using the honest-range-backed choice-between-lotteries may motivate voters to vote well on the range ballot, better than a small chance that the backup wins, with the probabilistic system. It may also be more manifestly "fair," because it is nonrandom.
If ε≥1 then step 5 can be erased.
If X happens (also) to be range voting, then we get a double range voting election-method where each ballot consists of one "sincere" and one "strategic" range-voting-style ballot. Observe that this voting method (and some of the others described here) all evade the Gibbard-Satterthwaite impossibility theorem, i.e. "accomplishing the impossible." This double range voting method with ε=1 will in practice (in terms of the winners it generates) very much resemble "range + (later) top-2-runoff," which is pretty good.
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