[EM] Compromise allocation of fair share (was Fair and Democratic versus Majority Rules)

[fsimmons at pcc.edu]

I just changed the thread name to reflect the current thrust more accurately.

Here's my best elaboration of the compromise allocation idea in the Party list PR setting:

1.  Anybody and everybody (corporations included, since the supreme court wants it that way) can nominate seat 
allocations to their hearts' content.  Let S be the set of all such nominated seat allocations.

2.  Each voter submits a ballot that either directly ranks all of the members of S, or else gives a well defined set of 
criteria for ordering them.  Call this set of ballots Beta.

3.  Each voter submits a function of the allocation percentages f(p1, p2, ...)=f(p) which (like a linear combination 
a1*p1+a2*p2+...) is homogeneous of degree one.  This just means that  f(t*p)=t*f(p).  Call this set of of functions Phi

4.  Use the members of Beta to eliminate all of the Pareto dominated allocations from S.   Let S' be the set of 
remaining nominatated allocations.

5.  The winning allocation is determined by the member p of S' that maximizes 
             the product (over f in Phi) of f(p) .

In other words, the respective parties P1, P2, ...get respective proportions p1 : p2 :  ...  of the seats.

With this method it turns out that each faction of size large enough to merit k seats can guarantee their favorite party 
P_j that many seats by submitting identical bullet ballots f(p)=p_j.

I'll outline a proof and give some examples in subsequent posts.

Forest



> 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 
> consensuscompromise. 
> 
> 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 
> difficultiesare 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 
> nominatedallocation 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 
> numbersaccording 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 
> substantialimprovement 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
> faction. 
> 
> 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:
> >
> 
> 
> ------------------------------
> 
> Message: 2
> Date: Fri, 10 Dec 2010 13:30:35 +0000
> From: Raph Frank 
> To: fsimmons at pcc.edu
> Cc: election-methods at lists.electorama.com
> Subject: Re: [EM] Fair and Democratic versus Majority Rules
> Message-ID:
> 
> Content-Type: text/plain; charset=ISO-8859-1
> 
> Another issue is the fact that the resulting legislature would 
> end up
> using majority rule for making decisions.
> 
> A legislature of
> 
> 60) A
> 0) C
> 40) B
> 
> gets the A faction almost all of its policies and the B faction 
> nothing.
> Replacing that by
> 
> 0) A
> 100) C
> 0) B
> 
> means that the A faction loses some of its policies, as C compromises.
> Thus the A faction will refuse.
> 
> Control of 60% of the legislature is better than 60% chance of control
> of 100% of the legislature.
> 
> I think to make it so the compromise works, you still need the random
> element. It is the threat that the "other-side" could win everything
> that causes compromise.
> 
> If power was actually shared in the legislature, then that issue goes
> away. For example, the rule could be that the national budget is
> shared equally between all legislators. A funding bill might require
> support from 1/3 of the legislature in addition to legislators willing
> to pledge a portion of their funding allocation. Another option would
> be to give legislators a finite number of votes and allow them cast
> more than 1 per motion.
> 
> Alternatively, you could introduce a small random element. The fall
> back could be standard list-PR, however 1/4 of the seats are reserved
> as bonus seats and given to one party. The odds of a party getting
> the bonus would be proportional to the number of votes it receives.
> 
> This means that a faction with 1/3 or more of the votes who wins the
> lottery will have a majority. They would get 1/3 of the 75% standard
> seats + 25% of the seats from the bonus, giving them more than half.
> 
> However, a minor faction would still need the support of other
> parties, even if they win the lottery (though their influence 
> would be
> greatly enhanced for that 1 term).
> 
> Another issue is that it would make parties much harder to 
> manage. A
> party couldn't offer potential legislators the potential of being
> careen politicians. This may or may not be a good thing. 
> However, it
> does mean that the degree of representativeness of the legislature
> would vary over time. Sometimes there would be wide representation
> and sometimes there would be narrow representation.
> 
> One option would be to make the bonus seats the only seats that are
> subject to the lottery/compromise system. This means that there is
> more stability.
> 
> The voting system could be
> 
> Each party submits a list
> 
> The votes would be
> 
> - voter marks at most 1 party as favorite
> - voter marks any number of parties as approved
> - voter marks any number of parties as acceptable
> 
> If the most approved party is acceptable to 90% of the voters, 
> then it
> is given the bonus seats.
> 
> Otherwise, a party is picked at random using the favorite votes and
> that party is given the bonus seats.
> 
> I haven't been keeping up to date on Jobst's latest single seat
> proposals, so there could be a better way to handle the specifics.
> 
> Another problem is one legislature taking decisions that bind later
> legislatures. For example, a legislature could increase the national
> debt or enter in long term agreements. A party which is 
> unlikely to
> have power after the next election is likely to try to take as many
> irreversible decisions as possible.
> 
> 
> ------------------------------
> 
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