Oh, additionally, von Neumann proved the minimax theorem, which is all that people need to talk about the things in poker they usually associate with game theory. In fact, when people call the jam/fold strategy "Nash" it's kind of funny, since Nash proved the existence of such equilibria for almost every game except this type! (which has been proved earlier).

EDIT: I know that "Nash equilibrium" applies to twoplayer zero-sum games also, I'm just saying. We don't call the identity on the disk a "Riemann map" even though it is.

It's too bad that your use of the sveltely economical "exploitive" was not similarly followed by those inclined to to the bloated -- but, alas, equally dictionary-approved -- "exploitative." That extra syllable, over the course of the next hundred millennia, will consume sufficient electrons to leave the internet a sere, useless utility.

Great work. Very useful.
I would say a "dominated strategy" is playing oop the most of your hands.

Nice instruction. Anyone will be happy to see the simple demonstration.

I only play/think about heads-up, but it strikes me that this is probably not true in 3+ player games... oops.

I'm new to this subforum and struggling while trying to read it because of unfamiliar terminology. Note that these are terms being used in this >theory< forum and thus I'm asking for clarification in this thread.

Anyway, going back to terminology, what do the following expressions mean:

toy game
villian (I understand it to be "the other player", but why such an evil-connotating term? Does it imply "another player that's playing badly", or "a player I loathe"?)

Hero/villain is a story telling convention - think of the old silent movies and vaudevillian plays: The essence of story telling is conflict, and the basic structure is a "hero" which does NOT mean a virtuous soul, but means the "one we should root for", and the villain, which is the rival of the hero.

In story telling, the hero could be evil, and the villain could be good. For instance, in the film "Heat" you cold say the HERO is the Robert De Niro character - a bank robber. The VILLAIN is the Al Pacino character, the police detective.

Or in a better example, Dog Day Afternoon, the HERO was Al Pacino as the bank robber, and the VILLAIN was Charles Durning as the police detective.

A toy game is basically just a small and simple game that we can solve exactly and otherwise play around with.

For example, the 1/2 street AKQ game: there are two players, the deck has just three cards: an ace, a king, and a queen. High card wins. Both players ante one unit. Hero can bet pot or check and go to showdown. If he bets, Villain can call or fold.

Super simple game, but its solution has some interesting properties (balanced, polarized betting range) that might give us some idea about how to play real poker...

Actually, it is true in zero-sum games. In an n-player game, let e_k be the expectation for an equilibrium strategy in position k, 1<=k<=n. Then

sum_k e_k=0 (due to the zero-sum property),

and thus

(1/n) sum_k e_k =0

which is the expectation of a player who plays each position with the same frequency.

I think the issue is that e_k isn't well defined here. Unlike in the heads-up case, in a 3+ player 0-sum game, there can be multiple equilibria with different expectations for each position.

Even if this were true (I don't see it, but I don't have a refutation neither; examples always appreciated), we assume that "all players play their equilibrium strategy," so we are at a (fixed) equilibrium. And for each such set of strategies the outcomes are defined.

Actually my reasoning only uses that there is a fixed strategy for each position.

It does, and there's not.

Ok, then the expression "their equilibrium strategies" does not make much sense in the multiway game. If some players choose strategies from one equilibrium, and others from another one with different expectations, then we're not in an equilibrium anymore and someone can exploit the situation.

Of course you could imagine that when I'm is on the button, then we play equilibrium A, and when I'm UTG we play equilibrium B. Then my argument doesn't work anymore.

Yea, that's pretty much the situation I had in mind, and I think that can happen in multi-handed play but not in heads-up. Or rather, if there are multiple equilibria in heads-up, they all have the same value.

My understanding of 3+-player poker games is pretty much all from MoP, Ch 29. It does a good job of explaining why they're theoretically interesting but doesn't give much in the way of practical advice or application.