Poker Theory: Maximal play versus Optimal play in poker

[JaredL]

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The strategy that is optimal for the game you're in is the one that maximizes your expected winnings given how your opponents are playing. You, like many others, are saying that some nonexploitable strategy is described as "optimal." It's not optimal in any sense, except in the case where your opponents happen to use a similar strategy and thus the game is being played in equilibrium.

Having taken several game theory and other microeconomic theory courses, and read way too many research articles in these areas, I have never come across a single game or micr theorist use the word optimal in the manner that you are.

Sorry to nitpick, it's just a peeve of mine. I think it came from Matt Matros, but have no idea where he got the notion. I suspect that he used the idea of "optimal versus maximal" or whatever to make himself look smarter.

[/ QUOTE ]I've never seen the term "optimal strategy" used to mean "maximal strategy" as you are using it. Any time I've ever seen the term used, it's meant "a prudential strategy in a two player zero sum game". Wikipedia and Mathworld both use it that way, and I'm pretty sure I've seen it used that way in books.

From Wikipedia (link): "Each player computes the probabilities so as to minimise the maximum expected point-loss independent of the opponent's strategy; this leads to a linear programming problem with a unique solution for each player. This minimax method can compute provably optimal strategies for all two-player zero-sum games."

From MathWorld's definition of "Minimax Theorem" (link): "The fundamental theorem of game theory which states that every finite, zero-sum, two-person game has optimal mixed strategies. It was proved by John von Neumann Eric Weisstein's World of Biography in 1928."

I agree the name is a bit misleading since if you know your opponent's strategy, an optimal strategy is usually not the best choice, but this usage of the term is standard in my experience.

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Note that in both cases, the strategies are best responses given what the opponents are doing. That's what makes them optimal.

[JackStrap]

I found reference in a thesis from CPRG

i think this quote represent JaredL toughts
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In the literature on game theory, a Nash equilibrium solution is often referred to as an optimal strategy. However, the adjective “optimal” is dangerously misleading when applied to a poker program, because there is an implication that an equilibrium strategy will perform better than any other possible solution. “Optimal”in the game theory sense has a specific technical meaning that is quite different, so the term equilibrium strategy is preferred.


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Algorithms and Assessment in Computer Poker.
Darse Billings, Ph.D. dissertation, 2006. 4.3
p.108 - Optimal versus Maximal Play

so use the term equilibrium strategy instead of optimum in my previous post...

[wax42]

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The strategy that is optimal for the game you're in is the one that maximizes your expected winnings given how your opponents are playing. You, like many others, are saying that some nonexploitable strategy is described as "optimal." It's not optimal in any sense, except in the case where your opponents happen to use a similar strategy and thus the game is being played in equilibrium.

Having taken several game theory and other microeconomic theory courses, and read way too many research articles in these areas, I have never come across a single game or micr theorist use the word optimal in the manner that you are.

Sorry to nitpick, it's just a peeve of mine. I think it came from Matt Matros, but have no idea where he got the notion. I suspect that he used the idea of "optimal versus maximal" or whatever to make himself look smarter.

[/ QUOTE ]I've never seen the term "optimal strategy" used to mean "maximal strategy" as you are using it. Any time I've ever seen the term used, it's meant "a prudential strategy in a two player zero sum game". Wikipedia and Mathworld both use it that way, and I'm pretty sure I've seen it used that way in books.

From Wikipedia (link): "Each player computes the probabilities so as to minimise the maximum expected point-loss independent of the opponent's strategy; this leads to a linear programming problem with a unique solution for each player. This minimax method can compute provably optimal strategies for all two-player zero-sum games."

From MathWorld's definition of "Minimax Theorem" (link): "The fundamental theorem of game theory which states that every finite, zero-sum, two-person game has optimal mixed strategies. It was proved by John von Neumann Eric Weisstein's World of Biography in 1928."

I agree the name is a bit misleading since if you know your opponent's strategy, an optimal strategy is usually not the best choice, but this usage of the term is standard in my experience.

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Note that in both cases, the strategies are best responses given what the opponents are doing. That's what makes them optimal.

[/ QUOTE ]It appears my examples aren't very good because the statements do make some sense under the optimal=maximal definition. Sorry about that. I'll see if I can find some clearer ones.

[JaredL]

OK, just to get this back on track. Relative to how you would play against expert opponents, if your opponents are:

you say -
If your opponents are too aggressive, you must be more tight

This tends to be true. Marginal hands become unplayable because you tend to have to pay more to play them.

- If your opponents are too loose, you must be more passive

This I disagree with. If they are too loose then the way to play tends to be more aggressive. Punish them for being too loose. Bet and raise when you have an edge.

If your opponents are too tight, you must play more aggressivly

This tends to be correct. You want to punish them by stealing their blinds early, and also put pressure on them late if they fold too often.

- if your opponents are too passive, you must be more loose

This is true. Against passive opponents you are able to get away with playing a lot more hands. This is the flip of the first one. You can get in cheaply with marginal hands on the first round of betting and will be able to draw cheaper on later rounds. Finally, when all the cards are out you aren't punished as much for having the second best hand.

[wax42]

It's not really that important, but here are some clearer examples of what I was talking about:

http://www.math.ucla.edu/~tom/Game_Theory/mat.pdf
p.16
"For a matrix game with m × n matrix A, if Player I uses the mixed strategy p = (p_1, ... , p_m) and Player II uses column j, Player I’s average payoff is sum(i=1 to i=m,p_i * a_ij). If V is the value of the game, an optimal strategy, p, for I is characterized by the property that Player I’s average payoff is at least V no matter what column j Player II uses"

http://www.math.wisc.edu/~swanson/in...ame_theory.pdf
p.10
"Perhaps the most important lesson to take from all of this analysis is the following:
You cannot win with the optimal strategy.
If you always play the optimal strategy, then as the opener you will have an EV of -1/18 no matter how your opponent plays, and as the dealer you will have an EV of 1/18 no matter how your opponent plays. In the long run, you will only be a break-even player. If you use the optimal strategy, your opponent cannot profit through superior play. But he also cannot suffer through inferior play. By playing “optimally” you have created a situation in which your opponent’s choices, good or bad, will have no effect on your EV."

http://www.cs.ualberta.ca/~darse/Papers/IJCAI03.pdf
p.5
"In a simple game like RoShamBo (also known as Rock-Paper-Scissors), playing the optimal strategy ensures a breakeven result, regardless of what the opponent does, and is therefore insufficient to defeat weak opponents, or to win a tournament ([Billings, 2000])."

I am also like 95% sure Sklansky uses "optimal strategy" in this way in the game theory chapter of "Theory of Poker", but I don't have that book with me right now.

[Abbaddabba]

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I continue to fail to understand why people so much like to use optimal/optimum to describe suboptimal behavior. It's certainly not coming from any game or economic theory.


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It's for people who want to artificially inflate their testicles.

Sounds real technical-like!

[JackStrap]

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- If your opponents are too loose, you must be more passive

This I disagree with. If they are too loose then the way to play tends
to be more aggressive. Punish them for being too loose. Bet and raise when you have an edge.


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i agree that value betting happen more often, one logical error is that with my arguments you must be more loose-passive against calling station what is not really a great maximal strategy

But, If you opponents are too loose there less bluffing , stealing and semi-bluffing because theses strategies loose values against loose opponents, in that sense i tough it can be applicable
againts maniacs(loose-aggressive) it's work because you should adopt a tight-passive style

i think i try too much to find relation between strategies... thanks for comments

[bbartlog]

The RoShamBo example illustrates the point well, but it's also a bit of a special case where the optimal strategy is guaranteed to do no better than random. In poker optimal play will not maximize your gains against opponents who play a non-optimal strategy, but it may still do pretty well.

For the most part I agree with JaredL's comments above (especially as regards being passive vs loose players - passivity is only good against particular kinds of maniacs). But in any case I would like to point out that when you talk about passive play, there are really three separate deficiencies that are getting lumped together, each requiring a different adjustment:

1) Not raising with strong hands. Against this kind of passivity, loosening up a little is good, since you can see more cards cheaply to try and complete your speculative hands.
2) Folding too much to aggression (this is actually 'weak' rather than 'passive' but I didn't want to add dimensions needlessly...). Obviously you need to be more aggressive yourself here.
3) Calling (neither folding nor raising) too much (the classic 'calling station'). You actually need to be less aggressive, in the sense that you only bet your actual made hands rather than pursuing a mixed strategy where some of your bets are bluffs.

Also, as regards opponents being too loose, I think that in general you can loosen up somewhat in proportion to their looseness. If someone plays any ace from UTG, you should be able to profitably play a range like AK thru AT against them, even though optimal strategy would dictate something more like AK and AQs only.

[grunta0]

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It's not really that important, but here are some clearer examples of what I was talking about:

http://www.math.ucla.edu/~tom/Game_Theory/mat.pdf
p.16
"For a matrix game with m × n matrix A, if Player I uses the mixed strategy p = (p_1, ... , p_m) and Player II uses column j, Player I’s average payoff is sum(i=1 to i=m,p_i * a_ij). If V is the value of the game, an optimal strategy, p, for I is characterized by the property that Player I’s average payoff is at least V no matter what column j Player II uses"

http://www.math.wisc.edu/~swanson/in...ame_theory.pdf
p.10
"Perhaps the most important lesson to take from all of this analysis is the following:
You cannot win with the optimal strategy.
If you always play the optimal strategy, then as the opener you will have an EV of -1/18 no matter how your opponent plays, and as the dealer you will have an EV of 1/18 no matter how your opponent plays. In the long run, you will only be a break-even player. If you use the optimal strategy, your opponent cannot profit through superior play. But he also cannot suffer through inferior play. By playing “optimally” you have created a situation in which your opponent’s choices, good or bad, will have no effect on your EV."

http://www.cs.ualberta.ca/~darse/Papers/IJCAI03.pdf
p.5
"In a simple game like RoShamBo (also known as Rock-Paper-Scissors), playing the optimal strategy ensures a breakeven result, regardless of what the opponent does, and is therefore insufficient to defeat weak opponents, or to win a tournament ([Billings, 2000])."

I am also like 95% sure Sklansky uses "optimal strategy" in this way in the game theory chapter of "Theory of Poker", but I don't have that book with me right now.

[/ QUOTE ] [img]/images/graemlins/ooo.gif[/img] [img]/images/graemlins/ooo.gif[/img]
The Theory of Poker . Page 185. " We can say,then, that if you come up with a bluffing strategy that makes your opponent do equally badly no matter how he plays, then you have an optimum strategy. "

[Jcrew]

To maximize EV, you need to play exploitive which from my general observation and level of abstraction, would seem to be playing slightly tighter than what your opponent is playing until you reach an inflection* point in tightness than you need to go into loose mode (full circle theory). Doing this would set up being exploited yourself.

*I would wildly guess that the inflection point would be close to an optimal strategy.