What is "Nash equilibrium" as it relates to NLHE?

[questions]

I've seen people make reference to that phrase several times now - most recently in the thread "will a perfect poker bot ever be created?" thread. I am unfamiliar with that term, and can someone please explain what it is, and what it means within the context of No Limit Hold Em and computer play? Thanks.

[_D&L_]

A nash equillibrium is simply where each player plays with the best response to his opponents strategy. Its an equillibrium, because no player has an incentive to deviate.


If you play rock, i play paper
If you play scissor, i play rock

goes around in circles....


but if you play 1/3 rock, 1/3 paper, 1/3 scissor
the best i can do is copy you.

Once each of us starts playing this strategy, neither of us has an incentive to deviate. Rock, paper, scissors is misleading because people may think Nash EQ means both players play the same strategy. This only occurs in Rock, Paper, Scissors because both players are acting simultaneously, with the same strategy set, and no assymetric information. They are "symmetric players" (my term) hence their strategies are symmetric.

Pocker is not so simple as rock, paper, scissors. There are rotating assymetries, assymetric information, and plenty of non-obvious dominated strategies (strategies that should never be played). Thus, even though the nash EQ breaks even with all other strategies in rock, paper, scissors, this does not happen in poker - far from it.

A nash EQ in a zero sum game will beat or break even with all other strategies (i mean after a full rotation at a table, their are positional advantages and a Nash EQ does not mean your just as strong in every position) . In poker, my own calcs is that a nash eq. will beat virtually all of the opposing strategy space; your chance of playing by luck a non-nash strategy that breaks even is virtual zero. You'll have to just take my word on that. I can prove it, but i'm not showing those proofs (or going to take the time to formally prove it).

[Paxinor]

@ DL: the positional advantage does not count in a nash equilibrium. there is basicly none, because the positional advantage is informational.

as you can see in the [0,1] games von neumann introduced in 1944 (those were as we know picked up by several pokerauthors) , position never matters.

there are some issues with impicit collusion though (in a 3+ player game) but let's just skip that as its kinda irellevant to the discussion right now.

@ question: basicly you will be not beatable if you play in a nash equilibrium which poker has (its been proven by nash himself) no matter what your opponents strategy is.

this strategy will be what we call in gametheory "mixed". meaning that you won't play a specific hand on a specific flop always similar. you might raise it 50% or might just call 50%. similar to roshambo where you dont always pick "rock"

it means that if a computer figures out a optimal strategy, he plays perfect poker. he will not maximize EV against every opponent but minimize his losses. he will be >= 0 against every player in the world.

but don't worry right now, its way to complicated to calculate the nash equilibrium even in limit HU poker

you might want to look in the "bot" thread in this forum there are further explanations...

[Gonso]

Questions,

A Nash Equilibria exists in situations where a given player can't improve his EV no matter what he does.

Let's we're playing a game. Player A does does one thing, and the Player B does his thing. If Player C winds up with the same result no matter what action he takes, then there's a NE. Most practical applications today for NE pertain to heads up jam-or-fold situations.

"Mathematics of Poker" covers this a little bit, and there have been a few good threads on it. You could look up the "SAGE system" from a Card Player article that was decent and tied in to this.

[_D&L_]

Look at page 18 of von neumann's thesis, located at this link: http://www.math.ucla.edu/~tom/papers/poker2.pdf

Can you tell me why it says that the "value of the game" for the first player is negative, and the value of the second player is the negative of this (hence positive)? And quoting the author, "We see that this game is in favor of player 2."

The last game is the only model where he allowed either player to initiate raising.

Even in von neumann's simple model, player 1s optimal strategy doesn't completely hide information from player 2, it just tries to hide as much as it can, and minimize the impact.

I should point out the information advantage is even greater in ring games. Are you really trying to contest that you can profitably play the same range of hands UTG as you do on the button? Or that you are going to be equally profitable in both positions?

I mean i could go into gametheory here, but its commonsense right?

[Paxinor]

nono we did not understand eachother right.

of course there is a positional advantage in multiple player games if you go into ranges because you have multiple opponents behind you so you need a tighter range, but this has nothing really to do with positional advantage we talk about on the flop for example when somebody likes to play IP because it gives you an advantage there.

you can of course open more lightly in late positions, because the game changes if an early player fold preflop

but: if an optimal player enters the pot in early position, you will not be able to exploit him without either folding too often or folding too less

but we should stick our discussions to HU, because the dynamics get so out of the imaginable that its hard to discuss it.

in HU there is basicly no positional advantage, because if the first player enters the pot, he will be neutral EV to any range that plays against him, even if he doesn't have position on the following streets.

of course the examle you made is not a correct proof of positional advantage because on page ten you see that one player has less options than another meaning its not a full street game because one guy can end the action directly by checking.

there are some asymetries i agree but its not comparable with the value of position in an exploitive game like today. meaning if you knew the optimal strategy you could probably play looser OOP than most people do today.

just that nobody confuses the positional advantage with the one sbrugby talks of when hes making a vid for cardrunners

[questions]

Thanks for the responses. [img]/images/graemlins/smile.gif[/img] Very heavy stuff, though. lol

[jstill]

[ QUOTE ]
In poker, my own calcs is that a nash eq. will beat virtually all of the opposing strategy space; your chance of playing by luck a non-nash strategy that breaks even is virtual zero.


[/ QUOTE ]

that last comment is especially stupid, most everyone out there is playing non nash poker trying to exploit the other players' weaknesses rather than trying to play unexploitably (becuz most players are bad and wouldnt correctly exploit unbalanced play on our part so its not an issue), the only players really focusing on playing unexploitably are HU lhe specialists and HU lhe bots mostly, even then good ones will immediately, and frequently deviate from it depending on their opponents tendencies (primarily how often they cr/ bluff semi bluff, raise the turn, raise preflop ect ect). In fact playing nash equilibrium poker all or most of the time would be the less profitable option, so to say you couldnt break even without it is completely erroneous unless everyone else is applying " nash optimal" strategy (which will probably never be the case in a game as complex as poker), in which case there really is no point in playing with the rake considered. Playing great poker means knowing when to focus on this unexploitable stuff would often be incorrect (less EV than something else) and when you should be thinking about something else, for example not folding a range of hands to a player thats a rock becuz folding vs most players cr'ing range on this board with your range of hands would be exploitable is actually incorrect (even if it is in accordance with what would be considered the unexploitably optimal play vs any opponent) since this particular opponent isnt cr'ing with a wide enough range to calldown profitably. U have to know when to think about nash in poker, and its really really rare tbh, against really good players who you play regularly I doubt most of us play in games where its necessary truthfully

[Paxinor]

jstill made a good point in general:

of course you should play exploitable because you should only play in games where players are so weak that you can exploit them.

so the use of nash equilibria at the moment are very limited (igonring that its to hard to calculate anyways today)

but there is definitly some attraction in the idea that you will make profit no matter how the opponent plays. especially if you are in situations where players are considered quite good but you don't have specific informations on them. or in games where players are really really good.

thats also why the SAGE system is quite popular. especially in low stack situations where you are not able to gather enough information about your opponents it is extremly useful to make sure you have at least a small if not bigger edge...

[indianaV8]

[ QUOTE ]
jstill made a good point in general:

of course you should play exploitable because you should only play in games where players are so weak that you can exploit them.

so the use of nash equilibria at the moment are very limited (ignoring that its to hard to calculate anyways today)

but there is definitly some attraction in the idea that you will make profit no matter how the opponent plays. especially if you are in situations where players are considered quite good but you don't have specific informations on them. or in games where players are really really good.

thats also why the SAGE system is quite popular. especially in low stack situations where you are not able to gather enough information about your opponents it is extremly useful to make sure you have at least a small if not bigger edge...

[/ QUOTE ]

Just to demonstrate your last point:

http://pokerai.org/pj2/images/stories/optVsExp.png

This is simple SS push/fold situation, you are on the small blind. You can see the EV for given push strategies - the different graphs represent different push strategies (optimal, push with top 8%, top 12% and so on). The Y axis is your EV and the X-axis is the call ranges (top X% of all hands) of your opponent.

Lessons learned? If you are against unknown, or very loose opponent it make sense to play optimal. If you are against tight opponent however, it make sense that you also play more tight than optimal, and in this case you can make upto factor 5 more profits (of course if you err on your reading, you will also loose quite some).