# [EM] Fair and Democratic versus Majority Rules

fsimmons at pcc.edu fsimmons at pcc.edu
Tue Dec 7 15:45:47 PST 2010

```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 interpretations
of 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 naïve 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 naïve 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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