re: [EM] Strong FBC

[Alex Small]

Demorep wrote:

>Will the voters be forced to state --
>this is my sincere vote --- [sincere vote]
>this is my insincere vote --- [insincere vote]

No.  Think of the election as a game, in the formal sense.  Your sincere
preference order represents your payoffs based on different outcomes, or at
least the relative order of which outcomes you prefer.

Your vote assigned by the computer, be it sincere or insincere, is your
strategy.  The computer assigns strategies to voters to find a Nash
equilibrium.

Although an election might have millions of voters, for a 3-candidate race
we can think of the game as having only 6 players.  If, for example, the
rules of the game were plurality voting, each player would have 3 possible
pure strategies.  For player 1 his pure strategies might be give 1 million
votes to candidate A, give one million votes to candidate B, or give 1
million votes to candidate C.  For another player, representing a different
bloc of voters with idential preference orders, the strategies might be
give 2 million votes to any of the three candidates.

Mixed strategies would be that bloc dividing its support among candidates.
I'm not sure if there's an incentive to do so, but since I'm using
terminology from game theory I might as well define the mixed strategies.

Now, the computer looks at the strategies and payoffs available to each
player and finds Nash equilibria, where each bloc of voters is following
its best strategies given the strategies of all other blocs.

Question:  Can we come up with a voting method such that you never have an
incentive to lie to the computer?  If so, then it doesn't matter if your
assigned strategy is insincere, we still satisfy strong FBC.

Why do I say that?

The concept of the computer assigning you a (possibly insincere) strategy
based on your sincere input is just one more step in a voting method.  We
can think of a voting method as a black box:  Input everybody's ballot, and
it outputs a winner.  If you have no disincentive to give sincere input,
then whatever magic and churning it goes through, assigning convoluted
insincere strategies, is all irrelevant.  Strong FBC is satisfied.

My hunch is that strong FBC cannot in general be satisfied.  For a given
set of players (with their associated payoffs) and available strategies,
there can be instances where there are at least two Nash equilibria, and
those 2 equilibria lead to different winners.  I don't know if there exists
a method in which insincere input cannot cause the computer to give at
least one person a better result.  If there is no such method then strong
FBC is impossible.

Maybe all I've done is make life more complicated without getting any
closer to an answer, but it may be that by putting an intermediate stage
between your sincere input and insincere strategy strong FBC can be
satisfied.  Something to think about...

Alex

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