They pick their numbers simultaneously, without knowing the choice of the other players. Similar to rock-paper-scissors.

If by "solution" you mean equilibrium, then as I have pointed out there are at least two, which consist of pure strategies.

By solution I mean is there a certain percentage of 0/1 A should pick to minimize his loss? Although this has nothing to do with poker, it's an interesting game... do you have a pointer to study more (or did you make it up on the fly) ?

I suppose it's an equilibrium... but there's no pure equilibrium _strategy_ that I see. A can't take the strategy of "pick the same number as the others", he can only take the strategy of "pick 0 x% of the time, pick 1 the rest". Same goes for B and C.
(Or do you allow for a strategy of "pick the number the majority picked the last time", i.e. is there memory? How does that work on the first round?)
BTW, since B and C both prefer picking 0, how does the game reach a state where everyone picks 1?

EDIT: hmm.... since B and C prefer 0, does that mean that there is exactly one pure equilibrium strategy for everyone, i.e. pick 0 100%?

In shallow 1 street 1CP (or any other poker I guess), BTN can say "I'll be going all in with X or better" and that'll set the equilibrium strategies for the other players. At least as far as I can see... is that correct?

I actually did

I'm not sure a solution in your sense exists, but I'll think about that some more.

An equilibrium always consists of strategies for all players, such that no player can improve his result by unilaterally changing his strategy. The constituting strategies could be called "equilibrium strategies".

The fact that an equilibrium strategy can't be exploited is specific to two-player games, so this doesn't work here. The problem is that, no matter what A does, B and C can collude against him.

No. If SB and BB can collude they can find a way to make BTN lose more than in the equilibrium.

Yes, but what is A's strategy? And the others? At least on the first round, it can't be "pick the same as the others". It's "x% 0, 100-x% 1". But how much each?
(I'm starting to think x=100 and the same for everyone, so it's one, not 2 equilibrium strategies for each.. but I haven't done the hard math. It's your game, tell me why would anyone pick something else )


You mean "collude" as in they play with common bankroll, and share the profits? That may be the case, not sure in a static 1CP game though. In a drawy game, maybe... if they can communicate during the hand, certainly.

But I said a couple of posts ago that I realize collusion can mess things up. Let's stick to a fair game for starters, 1 player and 1 bankroll per hand... it's an easier starting point.

Maybe this should be moved to a separate thread...

I thought it was interesting and relevant as it really gets at the question of what are the properties of an equilibrium and what is a solution concept in the case of 3+ player games.

Anyway, here's the new thread for anyone interested:

http://forumserver.twoplustwo.com/15...ibria-1292631/

I think that the terminology should be "GTO" for the base case, trivial, non-iterative solution to the optimum strategy, and the non trivial solution should be called the Nash equilibrium strategy. The reason I like this distinction is because it can then be compared to other games. All games have a GTO but only games like poker and prisoners dilemma have nash solutions.

I agree (ok you may call me partial), but i'd still actually see this thread in a cleaner state, it's a sticky and all. I.e. wouldn't mind if more of the stuff was moved.

The misinformation like "All games have a GTO but only games like poker and prisoners dilemma have nash solutions." should be cleaned out, imo.

one of the reasons why this forum is going to hell in a handbasket is because the strategies in the stickies are utter garbage

pure strategy - the purely mathematical optimum strategy known by both players...in poker, it's how the action would play out if no one ever bluffed, slowplayed, or overbet

mixed strategy - the use of counter strategies such as bluffs, slowplays, overbets and underbets to exploit the pure strategy

dominated strategy - usually used in reference to non zero sum games so doesn't apply to poker in general; however, there are certain post flop situations that can be considered a dominated strategy in the strict sense

unexploitable - pure strategies that can be proven to show an immediate profit, like raising AA pre-flop

One reason this forum goes to hell is that some muppets talk about things they don't have a clue about, and their nonsense goes unchallenged. Pure and mixed strategies have specific meanings in Game Theory, and those have nothing to do with you wrote there.

whatever, I hate this forum anyway, hope I get banned, but I'd like to get something useful out of it instead of Rock Paper Scissor strategy

Nash Equilibrium - a mathematical modification to the pure strategy equities, taking into account optimum counter strategies

Optimization - the plays necessary in poker needed to simultaneously maximize reward and minimize risk

Well, I don't see how misinformation can be useful, but knock yourself out. BTW, if you want to get banned you can probably just ask a moderator.

It's more fun to make fun of people who don't have a clue because I suck at RSP

I mean seriously this is flat out wrong and it's stickied

equity: the percentage of the total pot expected to be won should the hand reach showdown

I agree that some of the formulations in the OP are suboptimal, but what you write is flat out wrong.

I'm a little confused by this notion, which I've heard here before. I have a book in my hands published in 1961 which seems to use "optimal" in exactly the same way MoP defines it, and what tends to be called "GTO" in this forum. Here are some quotes:

[Games of Strategy: Theory and Applications, Melvin Dresher]

as far as I can tell, this usage goes back pretty much to the beginning of game theory. Is there something that MoP added to the word that I'm missing?

The text from OP that you quote is actually Rusty's addition after the discussion regarding the word 'optimal' ITT. But yea, the word is definitely used in the literature in both ways (i.e. to mean equilibrium and to mean maximally exploitative), and it definitely causes confusion. Personally, I've just stopped using the word when writing about poker theory.

So what do you use?

I use equilibrium, Nash equilibrium, GTO, unexploitable, and sometimes minimax pretty much interchangeably, at least for 2-player games, to refer to the one concept, and best response or maximally exploitative to refer to the other.

Let me comment just a little on this terminology thing again, and make some points that might help clarify why people get so confused.

The following is a list of strategies that I think we would generally like to give some kind of special name to:

(headsup poker between X and Y)

1) Let x be X's fixed strategy. Now Y has a simple optimization problem in pure strategies to maximize his expectation against x. (and vice versa if we hold Y's strategy fixed).

2) By the minimax theorem, there exists a game value V and mixed strategies x and y for X and Y, respectively, such X can do no better against fixed y than -V and Y can do no better than fixed x than +V. Finding these strategies is generally a complicated optimization problem.

(poker between more than 2 players)

3) If we hold all the other players' strategies fixed, we again have a simple optimization problem to maximize expectation. So a strategy comes from that.

4) By Nash's theorem, there exist at least one set of strategies for each player such that if the players all play these strategies, no individual player can improve his expectation unilaterally by deviating.

Now, what names should we give to these?

First off, there is a clear term in game theory for #4, which is a "Nash equilibrium." I do not believe that this usage is controversial.

AFAICT, economists (and the people who took a game theory class or something) seem to use "optimal" to mean "the result of optimization." So #1 and #3 are "optimal." This implies that there is no particular "optimal strategy," as it depends on your opponent's fixed strategy. Now #2 constitutes a Nash equilibrium, so they would probably call #2 "equilibrium" or possibly "minimax."

Math and CS people, on the other hand, are disinclined to call a strategy "optimal" simply because it is the result of a simple optimization. Further, the strategies in #2 are specially optimal - they are the result of the optimization against a rational opponent who will adapt to your strategy. Hence the use of "optimal" to describe #2 is common in the literature, especially the literature about the complex optimization problem of finding these strategies in zero-sum two-player games.

Now I have in the past expressed mild regret at using "optimal" and not "equilibrium" in MoP. That may not square with what I am about to write, but oh well. I now prefer the following terminology:

#1 and #3 are "maximally exploitive" strategies.
#2 are "optimal" strategies.
#4 are "Nash equilibrium" strategies.

The thing that this does is draw a stark contrast between the minimax strategies in two player games, which are really good*, and Nash equilibrium strategies in games of more than two players, which could be pretty bad, and are at best unreliable in terms of equity guarantees.

I would restrict my use of the "GTO" terminology to discussions with people who may not understand that the term "optimal" is being used in a technical sense, like any other term of art.
[*] If your response to this is "but not as good as human exploitation," you should go home and rethink your life.

I agree with everything you said but #2 is a subset of #4 so it cannot be technically incorrect to use the term "Nash Eq." in a two player game as well.

Regarding economists vs math/cs, why not just use the three letter acronym "GTO" by default (to refer to #2) and give no formal definition to "optimal". The word is thrown around so loosely now that imo, it's beyond redemption.

LOL. What do you mean by this? I think I get it but not entirely sure ...

Yes, of course it's not technically incorrect. None of this is like "correct" or "incorrect," -- it's just deciding what's best descriptive.

I just meant that optimal strategies are really strong, and there is always a decent subset of people who think that they are kind of breakevenish and win a little against bad players but fall far short of exploitive strategies against fish. I think this is generally false.

Makes sense.

Rest assured, those kinds of people aren't really the ones that frequent the theory forums all that often.

I think that this is a smart choice. (Nash) equilibrium and (maximally) exploitive cover all the optima we're interested in and make clear which one we mean, whereas in Jerrod's post, options 1 through 3 can be reasonably called "optimal".

However you seem to suggest that this does not hold for equilibrium strategies in multiplayer games. A few strong players appear to disagree with that.