[EM] Manipulation Resistant Voting

Susan Simmons suzerainsimmons at outlook.com
Tue Jul 20 16:34:37 PDT 2021

So the optimum strategy winner of a top-down ("choose a branch") binary tree structured election is also the sincere winner of the bottom up election based on the same tree.

And since a sincere CW will always win under sincere voting in the bottom-up case, the sincere CW (when there is one) will always be the optimum strategy winner in the top down case.

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-------- Original message --------
From: Susan Simmons <suzerainsimmons at outlook.com>
Date: 7/20/21 3:21 PM (GMT-08:00)
To: Kristofer Munsterhjelm <km_elmet at t-online.de>, robert bristow-johnson <rbj at audioimagination.com>, election-methods at lists.electorama.com
Subject: Re: [EM] Manipulation Resistant Voting

Here's the key: there are two ways to use binary trees for binary decision voting: bottom up and top down.

The meaning of honest/sincere voting is perfectly clear in the bottom up context, but the incentive for insicere voting is generally unavoidable.

Given the same tree, the optimum rational top-down strategy is to always vote for the branch whose expected outcome is greater.

Under perfect information with all rational players this top down optimum strategy solution is unique and sure ... the bottom-down winner will be the same candidate as the sincere bottom-up winner!

This can be proven recursively ... if it is true for both branches from the root node, it will be true for the entire tree. And (initial condition) it is obviously true for a subtree with only two leaves (candidates). These two facts are the only necessary ingredients for an inductive/recursive proof.  (Induction on the depth of the sub tree is another form of a recursive proof.)

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-------- Original message --------
From: Kristofer Munsterhjelm <km_elmet at t-online.de>
Date: 7/18/21 1:48 AM (GMT-08:00)
To: robert bristow-johnson <rbj at audioimagination.com>, Susan Simmons <suzerainsimmons at outlook.com>, election-methods at lists.electorama.com
Subject: Re: [EM] Manipulation Resistant Voting

On 7/18/21 5:30 AM, robert bristow-johnson wrote:
>> On 07/17/2021 5:12 PM Susan Simmons <suzerainsimmons at outlook.com> wrote:
>> ...
>> The Gibbard–Satterthwaite theorem states roughly that every
>> deterministic voting rule is manipulable, except possibly in two cases:
>> if there is a distinguished voter who has a dictatorial power, or if the
>> rule limits the possible outcomes to two options only.
> Could someone demonstrate here how, well outside a cycle, an
> insincere  vote can bring in a tactical advantage with a Condorcet rule?
> Say when would it be advantageous to bump your Number 2 to Number 1?
> Or when would it be advantageous to bury your Number 2?
> And without going anywhere near a cycle.

There are two cases where it would be beneficial to do strategy.

Number one is when there is currently a CW, but a faction can alter its
votes to create a cycle. Then it's beneficial if they prefer the cycle
tiebreaker winner to the CW. (Or vice versa, for that matter)

Number two is where there is a cycle and the tiebreaker itself is
vulnerable to strategy.

If the voters are constrained so that they can only submit ballots which
in aggregate makes a CW, then every Condorcet method passes IIA (since
if the CW is removed, it's not an irrelevant candidate, and if someone
else is removed, the CW remains the CW). I think, though I'm not sure,
that this also makes it strategy-proof.

My point, though, is that you don't just have strategy behavior inside
the cycle domain, you also have strategy by deliberately pushing the
method into (or out of) a cycle.

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