Alex--
You've answered my question: You really don't have anything better to do.
You said:
I think the reason why I called it "vague" is that at some point in that thread the subject of debate was Nash equilibrium, a very particular type of equilibrium. Due to the way the discussion had been going, I assumed that you were trying to define Nash equilibrium for voting, instead of some other type of equilibrium. That was my mistake.
I reply:
Oh I forgive you for that, Alex. Your mistake is excusable, considering that I call it voting Nash equilibrium, and repeatedly point out that it's the obvioius generalization of Nash equilibrium from the case where only one individual changes their strategy. :-)
In fact I still call it that. And it is the obvious generalization of Nash equilibrium from the case where only one individual changes their strategy. So it would be perfectly excusable even if you still continuied to make your mistake.
You continued:
The great thing about Nash equilibrium is that if everything in your game is suitably specified (and the criteria aren't all that strict) then at least one Nash equilibrium is guaranteed to exist. Your equilibrium obviously isn't a Nash equilibrium.
I reply:
Is supposed to be some connection between those two statements?
It's great that Nash equilibrium has that great thing about it when only one person changes their strategy. So your wishful thinking about what would be great, if it were true under very different conditions, is your basis for decreeing what can and cannot be named as a generalization of Nash equilibrium?
But things are actually different when a set of individuals change their strategy. Most people wouldn't expect the properties to all remain the same.
:
There now, I've cleared up another mistake for you.
Nash spoke of outcomes that no one individual can improve on for himself.
On EM, we spoke of outcomes that no set of individuals can improve on for themselves.
That is the obvious generalization of Nash equilibrium from the case where only one individual changes their strategy to the case where a set of individuals change their strategy.
And let me correct still another mistake of yours. You keep calling voting Nash equilibrium "Mike's equilibrium", "Mike's definition", or "Your equilibrium". One mighit have thought that I'd made it clear in previous postings that I wasn't the first to proposes that Nash equilibrium definition on EM.
You continued:
(see my example with Approval Voting and a Condorcet cycle)
Also, I have acknowledged that my formulation of Nash equilibrium for voting is not the only valid way to define a Nash equilibrium.
I reply:
...but you seem to havea forgotten that acknoledgement since then . You continued:
I have always pointed out that you could treat individual voters as players instead of factions of voters. I have gone on to argue, however, that such a treatment is not particularly useful, and that my definition of the players is the more useful one. That's all.
I reply:
Are you aware that that was discussed on EM years ago?
Anyway, even if we call a voter a player, that's useful as long as we speak of whether a set of plalyers can improve the outcome for themselves. A set of players is a generalization of a player. I didn't say that voting Nash equilibrium is identical to Nash's definition, only that it is a generalization of it.
Or, if you prefer, you can call a player any set of voters who change their vote attempting to all improve the outcom for themselves.
The word "player" isn't mentioned in my definition. But if Nash said "player", then you can take your pick of those two player-meanings that I've just suggested.
You prefer to add a definition that wasn't part of Nash's wording, the definition of a player as a set of voters who share the same preferences. That unnecessaly adds a stipulation that puts your definition farther from what Nash was talking about. There's nothing wrong with inventing, as long as you realize that that's what you're doing.
You continuied:
Anyway, I apologize for misunderstanding and thinking that you were trying to define a Nash equilibrium.
I reply:
That's ok. Your your error is entirely excusable, , considering what I've been saying all this time :-)
You continued:
I recognize that you were defining a different type of equilibrium.
I reply:
I do believe he's got it. Yes, we on EM defined an equilibrium that is for voting, for when there are many individuals, whole sets of whom can change their strategy, rather than just one. That's different from when we're talking about only one individual changing their strategy.
You continued:
Also, as to whether or not approval strategy depends on cardinal preferences in addition to ordinal preferences:
I completely agree that in the case of incomplete information it is necessary to factor in your utilities when deciding how to vote in an approval election. The equilibrium that I was defining, however, referred only to hindsight.
I reply:
No, that statement in your posting wasn't about an equilibrium referring to hindsight--You said that cardinal "preferences" weren't a factor in Approval strategy, and now you're trying to contort your way out of that mis-statement.
And here you've been doing so well so far, acknowledging your errors, but now you don't want to.
Mike Ossipoff
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