[EM] Fair and Democratic versus Majority Rules

fsimmons at pcc.edu fsimmons at pcc.edu
Thu Dec 9 16:27:06 PST 2010

So far we have established a formal analogy between lotteries (i.e. allocations
of probability among the alternatives) in stochastic single winner methods and
allocations of seats to parties in deterministic list PR methods.

We left off with the promise that Jobst's solutions to the defection problem in
the single winner lottery setting, when transferred (mutatis mutandis) to the
multi-winner setting, would (without sacrificing determinism) revolutionize the
world of list PR methods. 

All of Jobst's recent lottery solutions are based on this fundamental insight: 
We should take the random favorite lottery F as a basic benchmark of democratic
fairness, and consider any lottery C unanimously preferred (even if only
"weakly") to F as a frosting-on-the-cake improvement.  In the context of the
example of our previous installement  F is the 60%A+40%B lottery, and C is the
100% C lottery.  As in this example, so in general;  because C is (at least
weakly) preferred to F by 100% of the voters it is considered a consensus

How do we take advantage of the relation between F and C?  Simply put, we use
the threat of "fall back to F" as an incentive to prevent defection from C.

The question is put to the voters.  Do you prefer the fall back F strictly above
the proposed compromise C?  If even one voter responds affirmatively, then the
fall back allocation is used.

In the in the lottery context F is the random ballot lottery.  In the list PR
context F is the standard list PR method (whether based on Webster, Jefferson,
or Hamilton, etc.)

This idea is so simple, it's like post-it notes; everybody is sure to say, "Why
didn't I think of that?"

But, as they say, "The devil is in the details."   The technical difficulties
are all in how (in general) to automatically find a good compromise allocation
(of probabilities or seats, as the case may be) by a process that is immune to
manipulation.  In the simple example given in our previous post, 100%C is the
obvious compromise.  

In the following example

50  A 100,  C1 90, C2 40, B 0
50  B 100,  C2 90, C1 40, A 0

it is likewise obvious that the best compromise allocation is  50%C1+50%C2.

How do we find these allocations automatically?

Jobst has proposed many nice ways of doing this. One of the more recent ones is
this (slightly adapted to the list PR setting to get whole numbers of seats):

(1) Each voter rates the competing parties on a range style ballot.

(2) Each voter (optionally) nominates an allocation of the seats.  This could
well be done by choosing from a published list of such allocations.

(3) Each of the nominated allocations is tested against the fall back allocation
F on each of the ballots.  Whenever the fall back allocation F is strictly
preferred over a nominated allocation on even one ballot, that nominated
allocation is eliminated.

(4) If no nomination survives the previous step, then the fall back allocation F
is used. Else ...

(5) Calculate the compromise allocation  C  by averaging all of the remaining
(i.e. uneliminated) nominations together and converting to whole numbers
according to Jefferson, Webster, or Hamilton (consistent with the fall back
allocation F).

(6) Pit the compromise allocation C head-to-head against F to make sure the
conversion to whole numbers has not destroyed the unanimous approval for C.  If
it passes this test, seats are allocated  according to C.  If not, then the fall
back allocation F is used.

Jobst has invented more sophisticated methods than this one, but it is easy to
see that even this simple approach is entirely adequate for substantial
improvement over basic list PR.  

In particular, it gives the optimal solution to our first example, and, in the
second example will also give the optimal compromise iff all members of each
faction nominate their favorite lottery among those acceptable to the other

This entails that the voters in the first faction nominate 80%C1+20%C2 and the
voters of the second faction nominate 20%C1+80%C2.  Averaging over all voters
yields C = 50%C1+50%C2, as desired. 

One practical improvement in this proposed solution is to reduce the consensus
requirement from unanimity to some other quota between 95 and 100 percent.  This
pragmatic concession imparts additional robustness to the method when the
assumptions of perfect rationality and perfect information are remote from reality.

Enough for today.  I'll let Jobst write the next installment next week.

----- Original Message ----- (from Forest)
> Dear EM List participants,
> When last I wrote on the topic "Fair and Democratic versus
> Majority Rules" my
> purpose was to set forth some of the advantages of using chance
> to advance the
> principles of fairness and democracy as a remedy for the Tyranny
> of the Majority
> problem in single winner elections.
> Immediately the thread got off into Proportional Representation
> as the solution
> to Tyranny of the Majority, so the context of single winner
> methods was
> forgotten. When I tried to get back on track most readers were
> doubtful about
> the advantages of "lotteries" for use in serious elections.
> Although Jobst
> Heitzig elocuently answered all objections, and gave some
> examples in which the
> almost sure lottery solution was clearly preferable to the
> majority favorite,
> the thread dwindled into oblivion.
> I would like to try to resurrect the thread by showing how
> single winner lottery
> techniques can lead to better deterministic multi-winner PR
> methods. I alluded
> to the analogy between deterministic multi-winner PR methods and
> single winner
> lottery methods in my original post on the thread, but nobody
> except Kristofer
> really picked up on it, and he was doubtful of the value of the
> analogy.
> I just realized that the problem was psychological.
> Psychologically it is
> better to show how the analogy can be used to improve
> deterministic PR methods
> (which most list participants already believe in) than to use
> the analogy to
> convince participants of the value of single winner lottery
> methods (for which
> there is a mental barrier).
> To see the precise nature of the analogy consider two possible
> interpretationsof Jobst's challenge scenario
> 60 A 100, C 80, B 0
> 40 B 100, C 70, A 0
> In the single winner lottery interpretation, A, B, and C
> represent the
> alternatives. The 80 next to C in the majority faction row of
> the preference
> schedule means that those voters would prefer C to the lottery
> 79%A+21%B, but
> would prefer the lottery 81%A+19%B to the sure election of
> alternative C.
> Expressed as a compund inequality this information looks like this:
> 79%A+21%B < 100%C < 81%A+19%B
> This compound inequality is the content of the assertion that
> 100%C ~ 80%A+20%B.
> Similarly, the 70 next to C in the minority faction row entails
> the following
> approximate equality:
> 100%C ~ 70%B+30%A
> In both cases we have 60%A+40%B < 100%C, because in the first
> faction
> 60%A+40%B < 80%A+20%B ~ 100%C ,
> and in the second faction we have
> 40%B+60%A < 70%B+30%A ~ 100%C .
> Now for the second interpretation:
> This time the context is a multi-winner Proportional
> Representation (PR)
> election. Now the letters A, B, and C represent parties, and
> the numbers next
> to them represent the confidence had by the voters in the
> respective factions
> that the indicated parties will represent their interests.
> This time the inequality 80%A+20%B~ 100%C means that the
> majority faction
> voters would rather have all representatives come from party C
> than for 79% of
> them from party A and 21% of them from party B, but would rather
> have 81% of the
> representatives from party A and 19% from party B than having
> 100% of them from
> party C.
> In this interpretation, the inequality 60%A+40%B < 100%C
> represents the fact
> that both factions would prefer all of the representatives to
> come from C over
> the alternative that 60% come from A and 40% from B.
> In this PR context, note that any extant party list system will
> almost surely
> result in 60% of the representatives coming from party A, and
> 40% of the
> representatives from party B and none from party C, even though
> every voter
> would much rather have all of the representatives come from
> party C.
> All na?ve attempts at overcoming this problem fail. For
> example, getting
> everybody together and saying, "Since 100%C is preferred by all
> of us to the
> standard party list system result, let's just all promise each
> other that we
> will vote for party C." If ballots are secret, and the voters
> of the minority
> faction are honest, but the voters of the majority faction are
> low on scruples,
> the result will be 60%A+40%C, which rewards the defecting
> majority faction
> while penalizing the honest and loyal minority faction.
> In fact, under list PR rules, the game theoretic optimal
> strategy for both
> factions is to defect.
> Is there a way around this?
> The answer is "yes," and we can thank Jobst Heitzig for most of
> the work and
> inspiration behind the methods that best solve the problem,
> because he is the
> one who has consistently held our feet to the fire on the
> analogous issue in the
> single winner lottery case.
> To see this correspondence let's return to the single winner lottery
> interpretation. There the analogous problem is that if voters
> are allowed to
> directly assign their shares of the lottery probabilities
> according to their
> desires, the resulting lottery will be the "random favorite"
> lottery 60%A+40%B
> in which alternatives A and B have the respective winning
> probabilities of 60%
> and 40%, with C getting none of the probability, even though
> every voter prefers
> the 100% C lottery over the random favorite lottery.
> As in the PR analogue, all na?ve attempts at overcoming this
> problem fail.
> Non-binding agreements fail in the same way because defection is
> the optimal
> strategy when voters (by secret ballot) get to decide which
> alternatives get
> their share of the probability.
> At this stage many EM list participants would say, "The obvious
> solution is to
> forget lotteries and use some form of Range, since C is the
> obvious Range winner."
> There are two problems with this approach:
> (1) Because of the Tyranny of the Majority problem, rational
> voters who are well
> informed about each others' preferences will not elect
> alternative C under the
> rules of Range.
> More importantly:
> (2) Even if there were a deterministic method that reliably
> elected C, that
> would not help us in the PR analogue.
> Fortunately, Jobst has had the vision to see the value of
> lottery solutions to
> the single winner fair-compromise problem, since these solutions
> do transfer
> directly across to the deterministic party list PR context.
> Ironically (in the single winner context), any of Jobst's proportional
> probability lottery methods would almost surely elect C under
> the same
> assumptions about rationality and information that forced the
> failure of
> alternative C in the deterministic Range setting.
> I say "ironically," because only by making "chance" an essential
> part of the
> method can we make "sure" that C is elected in the single winner
> setting.
> Before continuing I want to emphasize that I am not proposing
> electing PR
> assemblies by use of chance. I am proposing that we use the
> analogy between
> stochastic single winner methods and deterministic multi-winner
> list PR to
> convert (mutatis mutandis) Jobst's single winner lottery methods
> into the best
> deterministic list PR methods the world has ever seen!
> To be continued:

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