[EM] SARA and a center squeeze scenario pair

[Jameson Quinn]

Let's say that the honest preferences are one of the following two
scenarios, and the voters don't know which:

35 or 35: A>B>C
10 or 20: B>A>C
15 or 05: B>C>A
40 or 40: C>B>A

Under SARA, the most naive/honest heuristic would probably be to rate the
middle preference "abstain", yielding the following:

35 or 35: A>>B>C
10 or 20: B>>A>C
15 or 05: B>>C>A
40 or 40: C>>B>A

Points: A70, B50, C80.
This gives a win to C or A, depending on which scenario is true.

Clearly, if all the B voters truncate, that's sufficient to elect B. But
let's say that the B voters are not actually that highly motivated to
strategize against their second choice, so they don't do that.

In scenario 2, the C voters can ensure that A can't win if 25 of them
switch to C>B>>A. This would give B 75 points, enough to beat A in scenario
2 but not enough to beat C in scenario 1.

But if the A voters anticipate this, then they have no hope of winning,
even in scenario 2; and thus, to avoid loss in scenario 1, all of them will
switch to A>B>>C. In that case, even if none of the C voters actually
strategize, B will still get at least 85; enough to win outright. So the
following ballots will be stable:

p1 or (1-p1)
35 or 35: A>B>>C at p2 or A>>B>C at (1-p2)
10 or 20: B>>A>C
15 or 05: B>>C>A
15 or 15: C>>B>A
25 or 25: C>B>>A at p3 or C>>B>A at (1-p3).

Results: (p1, p2, p3: result)
n,n,n: A
y,n,n: C
n,y,n: B
y,y,n: B
n,n,y: B
y,n,y: C
n,y,y: B
y,y,y: B

So if the payoff for B is 0, the A voters face a choice between a payoff of
Aa(1-p1)(1-p3)-Ac(p1)(p3). Insofar as B is over 50% of the utility of A, so
that Aa (the benefit they get from A winning) is less than Ac (the penalty
for C winning), the C voters must leave p3 less than .5 if they are to
tempt the A voters to not to give B points. Yet the payoff for the C voters
is strictly better the higher p3 is; leaving p low enough to tempt the A
voters not to give B points is dominated. Thus, (p2=p3=1 and so B wins with
certainty) is the subgame-perfect equilibrium.

If you're looking for a "trembling hand" equilibrium, B will only lose
insofar as the hands are very trembly and/or B is the score loser. This
seems to me to be a good outcome.
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