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

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.

Q.E.D.

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