[EM] more on lotteries

[Forest Simmons]

I'm still intrigued by the idea of electing lotteries for choosing 
candidates.  Here's an example of where they might come in useful:

Suppose that "true" preferences are

45 A>>C>B
30 B>C>A
25 C>A>B.

Then C is the Condorcet Winner, but the A faction, not liking C all that 
much, has an incentive to vote B>C instead of C>B, since then there would 
be a cycle which most methods would resolve by giving the win to A.

But suppose that we are using lotteries and the ballots are marked with 
the true preferences (including the ">>" mark to indicate that C is closer 
to B than to A for members of the first faction).

We compare the lottery C'=(.5, .5, 0) with the lottery C=(0,0,1):

The C' lottery has higher expectation than C for 45 of the voters, and 
lower expectation for 25 of the voters.

The B' lottery has greater expectation than B for 70 versus 30 voters.

And A beats A' 45 to 30.

The B' lottery has the highest approval relative to its opposite.

If we use this lottery B' to pick the winner, candidates A and C have 
equal chances of being chosen.

Now suppose that the first faction (of 45 voters) decides to vote 
(insincerely) A>B>C .

Then (according to the ballots) B beats B' 30 to 25, C' beats C 45 to 25, 
and A beats A' 45 to 30.

This time C' is the lottery winner.  This lottery chooses A or B with 
equal probability.

Now which result would the 45 A supporters rather have?

Would they rather have the result B' of their sincere ballots or the 
result C' of their insincere ballots?

Both of these give the same chance for A, but the sincere result gives a 
higher chance to C, their second choice, and no chance to B, their last 
choice.

So by using lotteries we have eliminated the incentive for the order 
reversal.

Since there is no incentive for the order reversal, the voters will vote 
sincerely, and the result will be B'.

Is this reasonable that A should have an equal chance with the Condorcet 
Winner?

I think so, considering that the largest faction rates C below average, 
and only the smallest faction rates C above average.

Also consider that A is the only candidate preferred over its opposite by 
a positive margin.

What if candidate C makes some concession to the A supporters immediately 
before the election, significantly raising their opinion of C, with the 
effect that their sincere ballots become

45 A>C>>B .

Then we would have B' beats B 70 to 30, A beats A' 45 to 30, and C beats 
C' 70 to 0.

This time the Condorcet winner C is the method winner with the new sincere 
ballots.

Would the members of the first faction prefer this result C to the result 
C' of their insincere vote (as calculated previously above)?

Well, evidently the 45 with (sincere) A>C>>B would rather have the sincere 
winner C, than the insincere winner C', so again there is no incentive to 
vote insincerely.

Does it make sense that with the increased support of C in the first 
faction that A should lose its chance of winning?

Yes, considering that now the CW candidate C is preferred over its 
opposite, and the largest faction now rates the CW above average, etc. 
The CW is now strong enough to hold its own without any defensive strategy 
on the part of the C supporters.

In other words the lottery approach (when done right) can eliminate the 
incentive for offensive strategy as well as the need for defensive 
strategy.

I haven't yet codified what is meant by "done right," but I'm working on 
it.

Any ideas for extension to more than three candidates would be 
appreciated.

Unfortunately I cannot just use "sprucing up" to reduce to the three 
candidate case, because then the ballot CW would always win, even when 
beaten pairwise by its opposite. (Imagine what would happen in the first 
sincere ballot case above, if it were spruced up.)

If the above example were conducted under some spruced up method, say 
Condorcet Lottery, then (with sincere ballots) C would win, and with
insincere ballots, candidates A, B, and C would each get one third 
probability.

How would the 45 voters with sincere preferences A>>C>B feel about this?

Let's say that ratings for A, C, and B are 1, x, and 0, respectively.

Then the question is which is greater (1+x+0)/3 or x ?

As long as x is less than 1/2, as indicated by the ">>", the
(1/3,1/3,1/3) lottery would be preferable.

So "Condorcet Lottery" encourages insincere voting in this case. [This 
was Bart's objection to Condorcet Lottery.]

Furthermore, candidate B who has 70 percent of the last place sincere 
votes shouldn't end up with a positive probability of winning as a result 
of somebody's insincere ballots.

So you can see why I am trying to do some other kind of lottery than 
"Condorcet Lottery," as well as why the method has to be applied before 
any sprucing up.  Sprucing up before applying a lottery technique will 
always end up more or less like "Condorcet Lottery" which isn't good 
enough by itself to remove the incentive for offensive strategy or the 
consequent need for defensive strategy.

I know that this isn't easy reading.  God bless anybody who believes it's 
worth thinking about in spite of my hurried exposition!

Forest