[EM] Andy's Question

Juho Laatu juho4880 at yahoo.co.uk
Mon Aug 1 02:44:20 PDT 2011

```One can model these questions also as requirements of the society. Different elections could have different targets. Maybe it is possible to give a list of possible parameters that can be used to determine what those needs are.

The society could try to maximize the sum of utilities, to to maximize the smallest utility and/or to keep all the utilities at the same level. One could also have some balanced mixture of such requirements.

If the election will decide which movies will be shown in television, and the method elects {A, B, C} (with the original votes below), then the AC voters will get 2 good movies to watch, and they will thus get more than the AD voters (1 movie). If the election picks toys for a kindergarten then AC voters must share toy A with 60 kids and toy C with 40 kids. Their expected utility can be estimated to be 1/60 + 1/40 while the utility of AD voters is 1/40. If the election picks hotels to sleep in for next night, then both AC and AD voters will get utility 1.

Also strategies tend to have an impact on what kind of methods societies use. Majority rule is a common approach. In the discussed example the latter voter could be an example of free riding.

With these considerations I mainly want to point out that it depends quite lot on the election in question how good choices {A, B, C} and {C, D, E} are. The needs of the society may thus determine what criteria we should respect.

The formulation of the "envy free" criterion that you presented is maybe not quite perfect yet.

> "If, when electing n winners, it is possible to divide the electorate into n groups of between one and two Droop quotas each, and each of these groups approves of a candidate that no other group approves of, then those n candidates should win."

Let's say that we want to pick two winners with the vote set below.

40 AC
50 BD

Does the criterion say that we should pick B and C? (and maybe the others too) Should the groups sum up to 100 or do 40 AB and 50 CD form a valid set of groups?

Are both {A, B} and {C, D} valid and required results?

Juho

On 31.7.2011, at 13.59, Kristofer Munsterhjelm wrote:

> Juho Laatu wrote:
>> Andy Jennings' question is a good question.
>> The original votes were
>> 20 AC
>> 20 AE
>> 20 BC
>> 20 BD
>> 20 BE
>> Let's decrease the support of A and B a bit (20 approvals reduced
>> from  both of them).
>
>> 20 C
>> 20 AE
>> 20 C
>> 20 BD
>> 20 BE
>> Would {A,B,C} be a good choice now? It is not good if reduction of
>> approvals makes A and B winners. And adding those reduced approvals
>> back shouldn't make A and B losers.
>
> {ABC} is not as obviously a bad choice as before, but if we want monotonicity, we're more or less forced to keep {CDE}. I think that {CDE} could still work, because each group gets an approved candidate. However, one of the 20: C groups could possibly have got better representation by voting for some other candidate in addition to C.
>
> The candidate assignment is like this:
>
> 20: C <- gets C
> 20: AD <- gets D
> 20: AE <- gets E
> 20: C <- gets C
> 20: BD <- gets D
> 20: BE <- gets E
>
> Thus, each group of 40 gets one candidate. We might even consider a (very limited) criterion for this sort of "envy free nature". It would go like this:
>
> "If, when electing n winners, it is possible to divide the electorate into n groups of between one and two Droop quotas each, and each of these groups approves of a candidate that no other group approves of, then those n candidates should win."
>
> In the example above, the group assignment would be:
>
> 40 C voters  (candidate = C)
> 40 *D voters (candidate = D)
> 40 *E voters (candidate = E),
>
> with n = 3. The Droop quota is 120/4 = 30, and these groups have sizes between 30 and 60 voters, so no problem there.
>
> In a sense, this is a more proportional version of Warren's representativeness criterion - I think that was the name - that if there's an assignment of candidates so that at least one of the candidates are approved by every voter, one should pick this assignment.
>
> The representativeness criterion, unadorned, is of limited use because it forces undesired outcomes in settings like:
>
> 999: AB
>  1:  E
>
> which then must elect AE, so a few other voters would defensively vote for other no-hopes to make representativeness inapplicable. The "between Droop quotas" blocks this sort of undesired outcome, and does so more the tighter the limit is. For instance, you might say "between a Droop quota and a Hare quota plus one, exclusive" and get a stronger version, but it would also be more specific and thus cover fewer cases.
>
> I also imagine it would be possible to generalize the criterion - e.g. if n/2 groups of appropriate size who each approve of at least two members, each has a subset of the set of candidates approved by that group, so that these subsets are disjoint between the groups, then the union of those subsets should win. But that gets really complex and covers only a few instances, and perhaps it's incompatible with the one-candidate case. For instance, you could have a 2-winner instance where groups of k approve of C, D, E, respectively, and groups of 2k approve of {FG} and {HI} respectively. I haven't checked if that's actually possible with Droop quotas though!
>
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