[EM] equilibria

[Alex Small]

(NOTE:  Please send future correspondence to my Yahoo address.)

James Green-Armytage said:
>  Alex gave an example which he claimed was an equilibrium for an
> electorate with no sincere CW, but I think I have contradicted him
> successfully: See below.

James-

First, EVERY game, if properly defined, has a Nash Equilibrium.  Find a case that doesn't, satisfy some mathematical fine print, and you can go to Stockholm to collect your Nobel Prize for proving John Nash wrong.

As to this example, it all depends on how you define the factions.  I don't like Mike's definition "No set of voters..." because if you define the players in the game too loosely then not only will there be no Nash equilibrium, but the game itself will be ill-defined and there really won't be much to say.

I defined the game in terms of factions with identical preferences.  This way, no matter what the situation is all of the people will have the exact same incentive or disincentive to change strategies.  Sure, some people in a faction might just slightly prefer candidate A to candidate B, while others strongly prefer.  But all of them will be able to agree that a strategic adjustment that elects candidate A instead of B is a good thing.

Before you ask "what about people in the real world who refuse to vote strategically?", keep in mind that this is game theory.  We can do almost anything we want as long as we clearly state the assumptions in our model.  You could model a situation where people care about sincere voting by producing a more complicated payoff matrix, in which people derive additional utility from voting sincerely.  That's a valid and interesting approach, and one that I've written about on this list in the context of CR strategy, but not the one being discussed right now.

>>Even if there's no Condorcet Winner there can still be situations where
>> at least some Nash equilibria involve all factions using pure
>> strategies. e.g. 
>>40 A>B>C----All 40 voters approve A and B 
>>35 C>A>B----All 35 voters approve C and A 
>>25 B>C>A----All 25 voters approve B only 
> [A wins]
>>(It's easy to verify that this is a Nash equilibrium.) 
> 
>  I'm not sure that it is. True, no single "faction" acting on its own
> can
> get a preferable result, but if the C>A>B and B>C>A factions fully
> cooperate, they can elect C instead of A. 

Anyway, in the particular case under examination, I agree that 2 players (B>C>A and C>A>B) have an incentive to cooperate and elect candidate C instead of A.  However, they are distinct players cooperating.  The subject of cooperative behavior comes up all the time in game theory, and I don't want to discourage you from studying it, but keep in mind that Nash equilibrium is DEFINED to refer to the incentives facing SINGLE PLAYERS.

Anyway, people can define whatever they want and talk about it, and people can discuss whatever they want.  If you want to talk about whether certain equilibria are robust against cooperative behavior then great.  But I have always understood a Nash equilibrium to be the following:

DEFINITION, Nash Equilibrium:  A situation in which no SINGLE PLAYER, ACTING UNILATERALLY can change the outcome to obtain greater utility than he obtains in the current situation.

Implicit in that definition is the notion that players should be defined unambiguously and in a manner that doesn't change depending on the strategies adopted.  In the case above, if we had a situation where B won then the B>C>A and C>A>B factions would no longer have an incentive to cooperate because one faction would prefer no other outcome to B winning, while another faction would prefer ANY other outcome to B winning.

So, the only question is "Who are the players?"  People can define the players to be whomever they want, but the notions of game theory will only be useful if those players have well-defined payoffs from each possible outcome.  Lumping 2 factions with different preferences into a single faction will not yield a faction with well-defined payoffs from EVERY situation.

When I study elections I choose to treat each group of voters with identical ORDINAL preferences as a single player.  The advantages of ordinal rather than cardinal preferences are

1)  It reduces the number of players to consider, keeping the game simple enough to analyze
2)  It still accurately encapsulates the incentives facing each faction when deciding on whether or not to change strategies.  EVERYBODY who prefers A to B will agree that a strategic adjustment changing the outconme from B to A is a good thing.  (See the above caveat on sincere vs. insincere voting.)

Some might say that individual voters should be the players in the analysis, but unless the margin is a single vote the situation will always be a Nash equilibrium because no SINGLE player can change the outcome.  You're probably thinking "yeah, but if a number of people change their strategies then the outcome could change, even if the margin is greater than one vote."  And that's exactly my point.

Anyway, please don't ask me to defend anybody else's definitions.

 

Alex

		
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