[EM] Interpreting Balinski's MJ words

[Kristofer Munsterhjelm]

On 01/03/2017 08:44 PM, steve bosworth wrote:
> To EM:
> 
> Please check the following [clarifications or corrections written within
> the square brackets].  Currently I see these suggestions as more clearly
> expressing B&L’s own intentions in the following two extracts from M.
> Balinski and R. Laraki (2011) /Majority Judgment/, MIT.  Please explain
> if you think I have misunderstood these paragraphs:
> 
> p.14:
> 
> Similar reasoning shows that the majority-grade mechanism is /group/
> 
> /strategy-proof-in-grading/.  A group of voters who share the same
> beliefs (e.g.
> 
> they belong to the same political party) has the same optimal strategy,
> namely, to
> 
> give to the candidates the grades it believes they merit. For if the
> group believed
> 
> that Royal merited better than /Good/, and all raised the grade they
> gave her,
> 
> her majority-gauge would remain the same (/p /does not change) [her
> majority-gauge would probably only be changed insignificantly when
> thousands are voting (/p would/ probably only be changed
> insignificantly)]. If all lowered
> 
> the grade they gave her, her majority-gauge would decrease (/q
> /increases), and
> 
> perhaps her majority-grade would be lowered (not their intent). If
> [instead] the group
> 
> believed that Royal merited worse than /Good/, and all lowered the grade
> they
> 
> gave her, her majority-gauge would remain the same (/q /does not change)
> [her majority-gauge would probably only be changed insignificantly when
> thousands are voting (q/would/ probably only be increased
> insignificantly)]. If [instead] all
> 
> raised the grade they gave her, her majority-gauge would increase (/p
> /increases),and perhaps her majority-grade would be raised (not their
> intent).

Basically, you're right.

Suppose that a group thinks Royal merits a Good. What B&L says is:

- We want to show that the optimal strategy for this group is to give
Royal a Good grade.

- Suppose for contradiction they give Royal another grade, and compare
this to what would happen if they gave Royal a Good.

- If the other grade they gave is better than Good, either nothing
changes (most likely when thousands are voting), or Royal gets a better
majority grade than Good (which is not what the group wants).

- If the other grade they gave is worse than Good, either nothing
changes (ditto), or Royal gets a worse majority grade than Good (which
is not what the group wants).

- So if the group wants Royal's grade to be Good, giving Royal any other
grade but Good only risks results the group doesn't want.

- Thus, grading honestly is optimal.

The point isn't that the voters are very unlikely to change the outcome,
but rather that even when they change the outcome by voting dishonestly,
they change it in a way that they don't benefit from.

> p.15:
> 
> One means by which it [MJ] resists [manipulation] is easy to explain.
> Take the example of Bayrou with a /Good/+ and Royal with a /Good/− (see
> table 1.4); their respective
> 
> majority-gauges are
> 
>  
> 
> Bayrou: (44.3%, /Good/, 30.6%) Royal: (39.4%, /Good/, 41.5%).
> 
>  
> 
> [Given these two majority-guages] How could a voter who graded Royal
> higher than Bayrou manipulate? By changing
> 
> the grades assigned to try to lower Bayrou’s majority-gauge and to raise
> 
> Royal’s majority-gauge. But the majority judgment is /partially
> strategy-proof-in-/
> 
> /ranking/: those voters who can [might marginally be able to] lower
> Bayrou’s majority-gauge cannot [significantly] raise
> 
> Royal’s, and those who can [might marginally be able to] raise Royal’s
> majority-gauge cannot [significantly] lower Bayrou’s.

Again, the point here isn't directly that the outcome doesn't often
change. (That the outcome doesn't often change is beneficial for other
reasons, some of which are related to strategy.)

The main point is instead:

Suppose some voter ranks Royal above Bayrou, like this:

Excellent | Good | Fair   | Poor | Reject
..........|.Royal|.Bayrou.|......|.......

Then if the voter wants to make Royal rather than Bayrou win, she can
push Royal up or Bayrou down.

But she can only push Bayrou down from where Bayrou is ranked on her
honest ballot, and can only push Royal up from where Royal is ranked on
her honest ballot.

And in effect, this makes only half the ballot available to do either.
If the voter has ranked Royal highly, she can push Bayrou down but can't
move Royal even higher. On the other hand, if the voter has ranked Royal
closer to the Reject end, she can push Royal higher (up to Excellent),
but can't push Bayrou very far down.

So a voter can only either exaggerate Royal (if she graded Royal low on
the original ballot) or exaggerate Bayrou (if she graded Bayrou high),
but not both at the same time.

The example voter above has graded Royal pretty high and so can only
exaggerate by one step - by moving Royal from Good to Excellent. On the
other hand, she can move Bayrou all the way from Fair to Reject.

> 1.4. For example, significant numbers of voters cannot contribute at all
> either
> 
> to raising Royal’s majority-gauge or to lowering Bayrou’s (28% of those who
> 
> graded Royal above Bayrou). Moreover, those who can manipulate have no
> 
> incentive to exaggerate very much in any case, for it does not pay to do
> so (a
> 
> more detailed analysis is given in chapter 19).

Here you have something that's more related to the fact that MJ is
robust. A large number of voters simply *can't* change Royal or Bayrou's
majority gauge. This further helps with the strategy resistance beyond
what they mentioned above.