Nash equilibrium for voting?

Richard Moore rmoore4 at cox.net
Mon Jul 8 23:08:41 PDT 2002

```Mike wrote:

> I know that Richard questioned the meaningfulness of that
> equilibrium, because it sounds like bloc voting.

I don't remember what I wrote about this. I might have questioned
its practical meaning, but certainly in a theoretical context,
like Alex's Small Voting Machine, it is useful (with the suggested
modification by Alex, because utilities are only needed for
probabilistic strategies).

But there are many ways you could define an election as a game,
depending on who you consider the players to be. And practically,
the players are defined by how well groups are able to coordinate
their strategies.

If V is the number of voters, and F is the number of distinct
factions (per Alex's definition), then you could have many
possibilities for the number of independent players, P:

If the voters are unable to coordinate their strategies at all,
then you have P = V.

(It's been pointed out that there would be a heck of a lot of
Nash equilibria with this many players, since in almost any
circumstance except a near tie, no single voter can change the
outcome. However, you could focus on those Nash equilibria that
are the most stable, in the sense of how large a factor would be
needed, if you exaggerated the power of a given voter by that
factor, to allow the voter to influence the outcome by acting
alone. I don't have a formal definition for this measure of
stability, and haven't really thought it through, but hopefully
you get the general idea.)

If each faction is 100% coordinated (100% of its members are
informed of the strategy and can be relied on to participate in
that strategy), then you have P = F (each faction is a player).

If several subgroups within each faction are internally coordinated,
but there is no coordination between the subgroups, then you have
F < P < V.

If two or more factions with similar but not identical rankings
can coordinate a strategy between themselves, that is better
for each than each faction can accomplish alone, those factions
would form a single player. In such a case you could have P < F.

If I understand Mike's paraphrase of Blake's definition, it
would apply to any possible combination. For example, two of the
subsets of faction A (A1 and A2) will team with subset B1 of
faction B to form a strategy, while group A3 teams with B2, and
at the same time each subset of faction C (C1, C2, and C3) forms
a different strategy. This would lead to a valid definition of a
Nash equilibrium for that configuration in that election.

I have no objection to the definition that produces exactly F
players, but simply want to emphasize that it is only one of a
multitude of ways the set of players can be defined. I don't
hold that it isn't a valid approach for certain theoretical
explorations, especially given its simplicity. I would only
object to the idea that this definition is useful in real-life
public elections, where many if not most voters will strategize
independently from the rest of their factions.

-- Richard

----