Please somebody give me a game where the other 9 players are all in blind every hand. OMFG that would be the juiciest game ever. I'd probably bring a couple rebuys.

Well obviously. The op just said only trying to beat you, so I assumed they only gain utility from beat you. In a cash game its obv diff. If all they wanna do is bust you tho, they should all get it in every hand as they'll bust you. They love increasing variance.

Looooool yea. Just like if I flip a coin twice I'm garenteed a heads. The point is by playing against n players all playing against you, the effective brs mean you get ruined so often. You get fcked by variance.

if everyone just jams and you close your eyes and call everytime wouldnt it be like sitting at a table and flip...

Flipping with 1/9 odds.

Yes, but why wouldn't you just wait for a hand?

Clearly in the scenario, your opponents do care if they lose money to you as well, not that they just have the objective to "bust you".

getting 9:1 on your money

Obv its nuetral ev. I'm talking about variance and how if its a situation where their goal is to get you out, there gonna get it in feather light. This is a stupid thread anyway. No one has actually answered ops post (including me it seems).

@Aarron: Can you explain your reasoning as to why you believe any change in play would be to the advantage of the group rather than the individual? I believe that the setup of the game the OP introduced is simply a completely different game from hold 'em from the individuals perspective as well as the groups perspective. Being that the games are totally different, I wouldn't rush to say that the advantage immediately goes to the group. If I were the individual I would certainly play different from normal vs all those sharks, so you would claim advantage goes to me as well?

To those recommending the groups strategy be to all shove every hand, I think this doesn't work: Hero can wait 200 hands until he has AA , and then call their shoved...does anyone know how AA holds versus 9 random hands? Must be more than 33%. So EV is (.33)(1000) + - (.66)100 or 170 bb. (assuming 100 bb starting stacks) . By the time you pick up AS you've blinded away 30 bb for a cool profit of 140 BB per 200 hands...better than OP asked for

I think OP has asked an interesting game theory question which is not obvious. To get an idea for the answer i would first ask the same question for the following 1 card poker game: single cards are dealt to players randomly. Players with highest card split the pot. A single blind is posted. game is limit, capped at 4 blinds. Betting round ensues. Each player may call, raise or fold at any time. All players except hero may cooperate but no signals are allowed. They are allowed to meet before hand to form a cooperative strategy. Can hero break even?

If you were to have some edge by playing as individual against a group, the group could always fall back to play as individuals and your edge would be lost, so there cannot be such advantage. To prove that the group has edge, we'd need to find only one scenario, where group has edge over the individual, because the group trivially has edge over individual as a whole by playing all other situations as individuals. Such situation could be multiway pot, where individual acts first and bets. Now, first group member to act can make a call justified by pot odds without a fear of bluff raise from the last to act member. With a little thought, you can probably find stronger examples, but the point is that edges don't need to be that huge in single situations, as those situations occur frequently in multiway pots.

If group members aren't bluff raising it becomes easier for us to put them on a tight range when they do raise in late position in multway pots. If the action goes call call call raise...hero? We have already picked up some dead money in the pot from group members who are presumably folding to the squeeze. We can tighten up our preflop calling range and pick up some dead money with our strongest hands. When villains act this straightforward (i.e. not bluff raising) it seems like hero can compensate by adjusting his pf calling range. Not sure how bluff could be exploited by villains. Do u have a specific example of multiway pots in mind?

You're right about the individual having no advantage since the group can fall back to individual strategy. I guess I was just saying that a group member playing differently from normal in some situation isn't a proof of the groups advantage...unless I'm missing something (which is always possible)

It is hard to think of any foolproof example, since calculating exact value for this kind of situations is next to impossible.

The answer however, should be obvious when we look at equilibrium strategies. If individual adjusts to collusion, he is playing suboptimal game against individuals, so he can be exploited by falling back to normal non-colluding strategy.

Game theoretically a players advantage is determined assuming that they have access to the other players strategy before the game begins. So actually hero knows what mixed collusion/non-collusion based strategy villains are using. He should adopt by a mixed strategy of his own. Postflop the game can get quite tricky...my intuition is that in a preflop only game where we're playing for high card only the colluders have no advantage. For nlhe as was already noted we only would need to find one particular situation where the colluders have an advantage, since they can play as individuals at all other times and break even. I'm not at all sure what this would look like

Well, not exactly. Players advantage comes from positive value of equilibrium strategy, which is essentially ignorant about opponent's strategy. Let's say we have strategy x, which is equilibrium strategy in individual game. Now, let (Y, y) be equilibrium for a game where one player plays individual strategy y and others play colluding collective strategy Y := (y_1, y_2, ...). Notice, that y_n are not necessarily identical because of the positions in relation to individual.

It seems clear, that playing individual game's equilibrium strategy x against Y has negative value and also y has negative value against all the other players in table playing x. Now, as (y, Y) is equilibrium, neither party (I think we can consider players playing Y as single player from game theoretical point of view) can unilaterally gain value by deviating, and players playing Y can switch to play x for positive value, also Y has to have positive collective value and thus negative value for y.

Does that make any sense?

Some rant about things you've said. Suppose there is an optimal individual poker strategy x. We ask if VILLAINS colluding can beat HERO. Let EV(y,Y) denote the expected value for HERO when he uses strategy y, and collective VILLANS use strategy Y=(y_1,y_2...,y_9). Let X=(x,x,...,x) denote the VILLAIN strategy where they each play optimal individual strategy x. Suppose that Y is the best VILLAIN strategy, which makes max{EV(y,Y): HERO strategies y} as small as possible. Now it's hero's turn to choose y so as to make EV(y,Y) as large as possible for this fixed strategy Y that VILLAINS have chosen. From the definition of NASH EQUILIBRIUM, the HERO gets to know Y in order to help him choose y.

I agree that EV(x,Y) <= 0, but i'm not sure if EV(x,Y) < 0. I also agree that EV(y,X) <= 0, but again am not sure if EV(y,X) < 0.

I guess I'm not sure why any of the things you claim are strictly positive or negative, and not allowed to be zero (or completely fair). This problem seems pretty complicated. Intuitively I feel like colluding VILLAINS have no advantage, but i couldn't say why i think that.

By the way, is it even provable that there exists a strategy x, so that all players playing x is in a nash equilibrium?? I think some theorem says that there exist nash equilibrium for poker, but is it known if there exist symmetric ones (that is all players playing the same strategy AND in equilibrium)?

I think you got this a little wrong. Playing equilibrium strategy, we don't get see what kind of strategy opponent plays and then choose the best response. We choose our actions so, that our value is maximized against rational opponent's best available strategy.

I think it is quite hard to formally show, that EV(x, Y) < 0 and EV(y, X) < 0, but I think it should quite clear from our previous discussion. There seems to be some colluding tricks (e.g. rarely bluff raising your team mate mentioned before), that need adaptations from strategy y. If we asssume, that there is any value in such adaptations, then EV(x, Y) < 0. Once again, no formal proof, but it seems obvious to me, that such adaptations (i.e. assuming opponent rarely bluff raises another opponent) also make the strategy losing in individual game, so EV(y, X) < 0.

Depends on how we define the game. Of course different positions play differently, and if we think about single hand games (these games have non-zero values for different players), where players are in different positions, then their strategy is different. If we think position as a parameter to poker game which is played for several hands, for example one orbit (this game has zero value for each player), then the game itself is symmetrical and of course therefore if x is equilibrium strategy for player a, then it is equilibrium strategy for player b also.

This was an odd question:

In a cash game you have negative expectation your best move is unquestionably to leave.

However, there are scenarios where you can get stuck in a game in which the other players view you as a player to be bullied and will effectively collude against you.

There are several basic strategies that can be adopted to maximise your chances. Assuming that you are at least as good a player as your opponents:

1 - change your table image, this requires engaging the other players in chat or talking. The purpose is to connect with the players in a different way. Talk about current events, sports or politics. Also you want to project the fact that you are a player they wouldn't want to be making decisions against. Although, if you repeatedly find yourself in these scenarios this could be tricky

2 - Identify the weakest player. This will be the player with shortest stack combined with ability. This is the player you want to play against. Wait for them to be engaged one-on-one with you before making a stand.

3 - Play incredibly tight. Do not play any hand in open play other than AA, KK or QQ
AK suited is no longer a good hand 8+ players!

4 - Move against Left Hand player. If people are colluding against you they cannot give you free flops. This means that the BB when you are SB will be continually raised by weaker hands than normal. So although, you are under more pressure, they will also be caught in the crossfire.

5 - keep players crippled not dead: Effectively you against the table play occurs when you are bullying the table. The ideal for this is players over whom you have 6x or more stack. If you were to go all-in blind heads-up every hand with this ratio you should still win. So do not try to eliminate ultra short stacks. Let the other players feed chips to that stack for you.

6 - if you become short stacked pick your opponent not your hand. This should be the next smallest stack. Now you are looking to eliminate them while minimising your risk. In this case it is no longer the hand odds but the tournament / lose all your chips odds.

7. In desparation - go up against as many players as possible in one hand.

8. slow down the play - in a tournament you want the blinds to go up.

I am not going to discuss the best multiplayer strategy against the one as i think that this might encourage unethical play. But scenarios with flat prizes such as qualifiers often cause over caution and if players give ultra short stacks free rides you can be put against a wall. The aims have to become:

A) Become the largest stack
B) get over 40% of the chips at the table.

The tactical situation is very similar to what is being discussed. Only that once you get A) the second stack will not play you - You can get them to fold AA even by All-in on every hand.

In a raked game it would be insane to stay seated - if you cannot bring yourself to leave without a profit your best odds are all-in on a 40% then leave if you win. But it still has a big negative equity.