[EM] Re: Election-methods Digest, Vol 5, Issue 26

Alex Small alex_small2002 at yahoo.com
Mon Nov 22 12:38:25 PST 2004

It's been a while since I made that post, but as I recall I felt that your definition was too vague because it contained the phrase "no set of voters" rather than the more restrictive phrase "No set of voters with identical ordinal preferences."  Using the phrase "no set of voters" rather than "no set of voters with identical preferences" is too vague and can lead to situations with no Nash equilbrium.  This would violate Nash's Nobel-winning discovery that (with suitable definitions) there are always equilibria.
Suppose that we use Approval Voting.  If we don't define the factions carefully, and we have a Condorcet Cycle, then it will always be the case that a majority of the voters prefer some candidate to the current winner, and acting in concert that majority could elect somebody that it prefers to the current winner simply by approving that preferred candidate and no other candidate.  (They might have to vote insincerely to do it, but the point is that they could do it.)
I'll give my definition of Nash equilibrium in elections, and a rationale for why I prefer that definition to various other possible definitions.
First, I define the players:

Player:  A set of voters with identical ordinal preferences is treated as a single player.  The voters in that set need not submit identical ballots, however.  A faction that does not vote en bloc would be analagous to a player following a "mixed strategy" in standard game theory, and a faction that votes as a bloc (all players submitting identical ballots) would be analagous to a player following a "pure strategy" in standard game theory.
Rationale:  First, treating each individual voter as a separate player is certainly a valid definition of the players.  However, the only way a single player could change the outcome of the election would be if the margin was a single vote.  Almost every situation would be a Nash equilibrium since there would be no incentive for a single voter to change his or her stratgey.  Analyzing from that perspective wouldn't lend itself to understanding situations where collective action changes the outcome of an election (e.g. the "spoiler effect").
Second, say that 2 people have identical _ordinal_ preferences (A>B>C) but different cardinal utilities (voter #1 assigns A, B, and C the respective utilities 10, 5, and 0 while voter #2 assigns A, B, and C the respective utilities 10, 7, 1).  The fact remains that both of these voters would benefit from a strategic adjustment that changes the winner from B to A, and both would be hurt by a strategic adjustment that changes the winner from B to C.  Both therefore face the same strategic incentives, and it is there
Definition of Nash equilibrium in elections:  If all of the players are voting with (pure or mixed) strategies such that no single player can change the election result to a candidate that the player prefers to the current winner, then we say that this situation is a Nash equilibrium.

Also, notice that at no point did I make reference to whether or not the voting method uses ranked ballots.  Even in non-ranked methods (e.g. Approval Voting) your strategic incentives depend only on your ordinal preferences, not your cardinal preferences.  If you prefer A to B, changing your strategy to elect A rather than B is always in your best interests, regardless of whether you think the difference between the candidates is minor or huge.  Well, some might quibble with that, but it's certainly true if voters have sufficient information to guide their strategies.  Anyway, Nash equilibrium is defined with hindsight in mind.  We assume that no player _could_ change the outcome to one that the player prefers, not that no player _knows_ it could could change the outcome.
Anyway, please accept my apologies if I was wrong about your definition, but that's how I recall the dialogue.  Above is my definition, and if it matches yours then I guess we agree on everything.

election-methods-electorama.com-request at electorama.com wrote:
Alex wrote:

I don't like Mike's definition "No set of voters..." because if you define 
the players in the game too loosely...

I reply:

I wasn't aware that a definition of "players in the game" appeared in my 
definition of (voting) Nash equilibrium, as we'd been using that term here.
Alex continued:

then not only will there be no Nash equilibrium

I Reply:

But there demonstrably are (voting) Nash equilibria, as I defined the term.

Alex continued:

, but the game itself will be ill-defined and there really won't be much to 

I reply

About what? I didn't define a game, either ill-defined or well-defined.
What should there be to say, other than the definiiton, and statements about 
what meets the definition and what doesn't.

I and others felt that it was useful to speak of outcomes that no one 
could improve on, individually or collectively.

If "group strategy equilibrium" is in wide use, to mean exactly what we've 
been calling a voting Nash equilibrium, then I'm not saying that I object to 
changing to that other term. But saying that there's another word for it 
isn't the same as saying that the term is incorrect.

Mike Ossipoff
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