Gaming Theory

[AnotherAddict]

I have read the T.O.P. book and it discusses gaming theory, can someone give me a practical example of using gaming theory for Hold-em?

[Jazza]

i love game theory, here (http://forumserver.twoplustwo.com/showflat.php?Cat=&Number=2347284&page=1&view=colla psed&sb=5&o=14&fpart=1) is something i posted, although it's not fully practical, so i'll give you a practical one here:

suppose that you are playing NL holdem, and are involved in a pot with one other opponent, you have AKo

you know your opponent well, and he knows you well, after the preflop, flop, and turn you have deduced that he would only and always play AA/KK/QQ like this, he knows you know that, and he has also figured out you have AKo, and you know this

the flop and turn were 2/images/graemlins/diamond.gif4/images/graemlins/club.gif8/images/graemlins/heart.gifK/images/graemlins/spade.gif

your opponent has $100 left of his stack, you have him covered, and there is $70 in the pot

now the river comes A/images/graemlins/spade.gif

first things first, what are the chances you have the best hand? it may look like it's 1/3, but it's not

there are 6 combos of QQ he can have, but only 1 combo of KK and one combo of AA he can have, so this means you have the best hand 3/4 of the time

he's first to act, he bet's amount X, you should call with probability P (raising is bad since he won't call with QQ)

P = 70/(70+X)

by calling with this frequency you make it EV neutral for him to bluff

if you call less than this frequency then he can bluff you off the pot more and increase his EV (which means decreasing your EV), and if you call more than this frequency he can bet only his AA and KK and get paid off more and increase his EV (which again means you decrease your EV)

so this is why this calling frequency is optimal

now suppose you're the player with AA/KK/QQ

we'll assume check-raising is not optimal, since i'm pretty sure the player with AKo should only ever call and not bet, so this means you should bet your AA/KK every time

but two questions are: how much should you bet, and how much should you bluff with QQ?

the optimal strategy is to always go all in for $100 with AA and KK, which will be 1/4 of the time, and bluff all in with QQ (1/4)*(10/17)=5/34 of the time

this means that 10/51 of the time you have QQ you should bluff all in

this means that when you go all in, the odds of you having AA/KK are 17 to 10, thus making it EV neutral for your opponent to call since he is getting 17 to 10 pot odds

if you bluff with QQ any less, your opponent can fold more and you will let him win more pots since you checked with QQ when you could have bluffed him off

if you bluff any more he can call more and you are bluffing off too much money

so in summary, the optimal strategy for the player with AA/KK/QQ is to go all in with AA/KK, and to bluff with QQ 10/51 of the time he has QQ

which makes the optimal strategy for the player with AKo to call the all in 7/17 of the time (since this is 70/(70+X), X=100)

the AKo player has a minimum EV of $10.294..., no matter how the his opponent plays

the AA/KK/QQ player has a minumum EV of -$10.294..., no matter how his opponent plays

these two strategies form a Nash Equilibrium (John Nash, like in the movie), a Nash Equilibrium is a strategy where if you fix all strategies but one persons, you can not alter that one persons strategy to increase his EV

in a heads up zero sum game there is one and only one Nash strategy (i think, this is something i came up with myself, can some one verify?)

one final note, these strategies are optimal only if your opponent plays perfectly, if your oppoenent is flawed there may be a better strategy

an analougy is paper rock scissors, with two players playing heads up the Nash Equilibrium is for both of them to pick each 1/3 of the time, which results in an EV of 0 for both players

but if your opponent is Bart Simpson who always picks rock, your optimal strategy is no longer to pick each 1/3 of the time, now you should pick paper

there is actually a paper rock scissors competition (http://www.cs.ualberta.ca/~darse/rsbpc.html) where people write programs to try and outthink each other

one last note:

when you know what your oppenent has, and you still could have a range of hands where some hands are better than your opponents hand, and some hands are worse than your opponents hand, the optimal strategy (against a perfect opponent) is to go all win with your hands that are better than his, and to bluff with hands worse than his enough so that the pot odds you give your oppoent are the same odds you have him beat

i've notice Spirit Rock/Mahatma on UB has often overbet the pot all in, and i'm wondering if he's doing this because he's narrowed down his opponent's hand range so much that he can now implement this strategy

[phish]

I'm not sure I follow all that mathematical treatise you just wrote, but this is how I interpret game theory to mean in practice at limit holdem:

Oftentimes at the bigger games, especially when it's shorthanded. You'll see something like this: a late position raise, the SB reraises, and a call. Then the SB bets and the original raiser will play back at him, etc. When all the cards are turned over, the SB may turn over something like K9s and the initial raiser may show AJ while the flop showed was Q83. This, imo, is game theory in practice. To explain:

Imo, game theory is playing in such a way that will not allow your opponent to have a BIG playing advantage over you. The general dictum is that the biggest suckers are people who call and chase too much. That may be true at the smaller full games where you essentially have to show a hand to win. But at bigger shorthanded (and ESPECIALLY headsup) games, the biggest suckers are people who play too weak and fold too easily.

Now with most hands (even good hands like AK or middle pairs), most of the time, you're not going to like your flop and when playing against 7 opponent can give it up. But if you played like that (giving it up when you hold AK and make no pair or hold 88 and flop comes Q93), you simply will get run over. Thus the appropriate approach then is that when you do decide to enter a pot, you are not looking to hit a flop and make your opponent pay off (tho that may happen), but you need to play in such a way that will maximize your chance of winning it. And, (this is where game theory comes in) to guard against that type of strategy employed by your opponent, you need to occaisionally show strength even even when you have nothing. By randomizing this playback, you will reduce his playing advantage to a minimal.

Going back to our prior example, the SB has a hand that he wants to play (whether you should play his hand in this particularly situation your choice). But instead of just calling and hoping to see a flop, he wants to squeeze out the BB and take the psychological lead to bet his opponent off the hand postflop. His opponent, employing game theory, will not automatically fold just cause he missed the flop. If he were to routinely do this (raise and fold like that), he will have no shot in the game. Hence game theory dictates that he should occaisionally play back even w/ nothing. (Of course he may run into a set of QQ sometimes, but in the long run it is much costier to not occaisionally play back). His willingness to hang tough will in turn intimidate his opponents from making too many moves against him. (You ALWAYS want your opponents to play more predictibly and passively).

I could elaborate further but you get the general idea. Some people call it metagame strategy or image or whatever. But I think this is game theory in practice without all the mathematical mumbo jumbo.

(Who knows, maybe I'm totally off the wall and this is not at all what people mean when they talk about game theory).

[RoundTower]

This is my first post here, but I do think this is something quite important.

Firstly let me say that I think Jazza's explanation of game theory is largely correct. However...

It is almost always NOT correct for you to bluff, or call on the river, with the game theory-predicted frequency. To see why you should note that if you are the AK player in this example, and you call with this frequency, your EV is $10.294 regardless of what strategy your opponent is following. Now either your opponent is following his "correct" game theory strategy or he isn't.

If he is, it makes no difference to your EV whether you fold or call. This means that it is very difficult to beat players who are playing perfectly.

If he isn't, on the other hand, you should be inclined to call more often or to fold more often, depending on the player.

This is your only edge in poker - exploiting your opponents' tendencies. (The most common tendency is that they play too loose...) You must deviate from game theory in order not just to maximize your win rate, but to win at all.

The exception to all this is when your opponents are better players than you are, and you are losing money. In this case game theory would provide a way for you to break even, but you shouldn't be in that game.

I hope I've made this reasonably clear; the post is too long already, and it could be quite controversial. Possibly 50 years from now limit hold'em will be completely "solved" so that computers can play perfectly according to game theory, but for now, I think the subject has little bearing on practical poker play.

[Jazza]

agreed, let the record show that i said:

[ QUOTE ]
one final note, these strategies are optimal only if your opponent plays perfectly, if your oppoenent is flawed there may be a better strategy

[/ QUOTE ]

the problem with adapting to your opponent, is that if adapt the wrong way or he re-adapts to you, you might end up making your EV worse

the paper rock scissors analougy is perfect, playing a perfect opponent you pick each 1/3, but this guaratees an EV of 0 even if your opponent is really bad

so if your opponent is bad and always picks rock, you can now always pick paper, but now you have to be carefull because your strategy can be exploited and you might be playing -EV if your opponent adapts, or you were mistaken that he always picks rock, just a word of caution

and if you are the player with AKo, and you play with the unexploitible strategy, your EV can increase from the minimum $10.294 if your opponent ever goes for a check-raise with AA/KK