Jameson Quinn suggested the following 1-parameter model about "one-sided strategy" in voting. Warren D. Smith then (April 2011) worked out the consequences of Quinn's model in closed form, finding a surprisingly elegantly simple result.
We had previously examined only the special ("unbiased," "honestly close election") case P=0 of this model. That special case turns out to behave unlike the P>0 cases. When P>0 we here argue median-based range voting is "twice as resistant to 1-sided strategy" as average-based. But when P=0 we previously had argued average-based range voting looked to have better resistance to 1-sided strategy than median-based! (Obviously, the P=0 special case is the most important point of this parameter space, but it seems dubious it should outweigh the whole rest of it.)
The model: Two candidates named "Gore" and "Bush." Large number of voters. Initially, each voter has honest score for each candidate that is an independent random-uniform real number in the real interval [0,1]. But now reverse a fraction P of the Bush>Gore votes, by applying the map x→1-x to its scores x. The result is a set of "honest" votes which, in net, favor Gore. These votes are described by a single parameter P with 0≤P≤1; the greater P is, the more pro-Gore the voters honestly are.
That situation will be our starting point. It has 100% honest voting. We will now consider the effect of pro-Bush 1-sided strategy.
This model implies there are a fraction (1-P)/2 Bush>Gore, and (1+P)/2 Gore>Bush, honest voters. The Gore>Bush voters have a triangular probability density of Gore scores: ProbDensity(x)=2x, with mean score 2/3 for Gore and mean=1/3 for Bush. The Bush>Gore voters have the same triangular probability density of Bush scores and mean score 2/3 for Bush and mean=1/3 for Gore.
Now suppose a fraction F, 0≤F≤1, of the Bush>Gore voters strategically exaggerate to Bush=1, Gore=0. We'll assume there is no counterstrategy by the Gore>Bush voters – they just stay honest.
THEOREM: In the above model with any P with 0<P<1, Gore wins with honest voting; but Bush wins with 1-sided exaggeration exactly when the fraction F of Bush>Gore voters who exaggerate obeys F>P/(1-P) for average-based range voting, and F>2P/(1-P) for median-based range voting. If P>1/2 and P>1/3 respectively, then Gore wins despite any amount of pro-Bush exaggeration. Approval voting (since it 'forces exaggeration') is entirely immune to 1-sided strategy, i.e. with it Gore wins, period.
PROOF: We first shall derive formulas for Bush & Gore's post-exaggeration median and average scores. (The pre-exaggeration results also will arise from our same formulas by taking F=0.) Note that there are three kinds A,B,C of voters, present in proportions
who respectively rate Gore=0, Gore=honest, and Gore=honest. The type B∪C voters (combined set) have a trapezoidal probability density of scores with the two trapezoid leg-heights being (1-F)(1-P) and (1+P), up to a constant multiplicative scaling factor you need to choose to make the total probability-mass be 1, i.e. "normalized," at the Gore=0 and Gore=1 sides respectively. Gore's median score is then m where
This simplifies to
a quadratic equation whose solution m is
This simplifies to
a quadratic equation with solution
(note the integrals in the denominators are there merely to provide the right normalization-constant for the trapezoidal probability density). This simplifies to
Finally, Bush's average score is
which simplifies to
Now we ask: what fraction F of Bush>Gore voters need to exaggerate in order to make Bush win? With averages, the answer is got by solving BushAverage=GoreAverage for F, with the very simple result
Note that if P>1/2 then average-based range voting becomes immune to strategy: no amount of exaggeration will suffice to make Bush win. I.e. this model yields such immunity when the honest ratio of Gore>Bush to Bush>Gore voters exceeds 3:1.
With medians, the answer is got by solving BushMedian=GoreMedian for F, with the amazingly simple result
Note that median-based range is thus exactly twice as resistant to 1-sided strategy as average-based range, in this model, in the sense that exactly twice as many exaggerators are needed to make Bush win (regardless of P for 0<P<1).
Median-based range voting becomes immune to strategy in this model when P>1/3, i.e. when the honest ratio of Gore>Bush to Bush>Gore voters exceeds 2:1.
Finally, for approval voting, the votes are "already exaggerated" to the endpoints of the allowed score range, so further exaggeration has no effect. Hence if Gore wins, he wins.