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Super Duper Basic Game Theory Concepts

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  1. #1
    spoonitnow's Avatar
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    Default Super Duper Basic Game Theory Concepts

    Introduction

    I'm writing this post because game theory is math and math scares people. I think it would be nice if everyone had a basic idea of what certain terms and expressions mean regarding game theory since it could seriously lower the number of misconceptions about this particular branch of mathematics.


    Contents

    1. What is game theory?
    2. Exploitive Play
    3. Game Theory Optimal
    4. Optimal Exploitive Strategy
    5. Adjusting
    6. Toy Games
    7. Half-Street Games
    8. [0,1] Games
    9. Game Theory Concerns for Micros


    1. What is game theory?

    Wikipedia gives us: Game theory attempts to mathematically capture behavior in strategic situations, in which an individual's success in making choices depends on the choices of others. Dictionary.com gives us: a mathematical theory that deals with strategies for maximizing gains and minimizing losses within prescribed constraints.

    2. Exploitive Play

    Exploitive play attempts to take advantage of certain opponent tendencies. If you were playing heads up, and your opponent open shoved 50bb with every single hand or shoved over you preflop if you didn't fold, it's pretty obvious that you could exploit this by calling with hands that were +EV against any two cards in those spots. There are less obvious things that we can exploit, the general theme of which is when our opponent takes a certain action too often or not often enough, or when our opponent's range has certain characteristics.

    3. Game Theory Optimal

    Game Theory Optimal play is a strategy that cannot be exploited, but it's not always the highest-scoring strategy against a given strategy. In paper-rock-scissors, optimal strategy is randomly choosing one of the options at a rate of 1/3 paper, 1/3 rock, and 1/3 scissors. If an opponent was playing 100% paper, optimal strategy would score lower than a strategy of playing 5% paper, 5% rock, and 90% scissors. In turn, our strategy becomes exploitable since we moved away from optimal strategy. In addition, game theory optimal strategy typically makes an opponent indifferent about his different options.

    4. Optimal Exploitive Strategy

    An Optimal Exploitive Strategy is the best performing strategy against a known opponent's strategy. In paper-rock-scissors, for example, the optimal exploitive strategy against someone who plays 100% rock is to play 100% paper. A different strategy that's exploitive but not optimally exploitive would be 90% paper and 10% rock.

    5. Adjusting

    Sometimes when we notice a tendency in an opponent, we can make an adjustment in our own strategy to exploit it. A common example in no-limit hold'em is a player who open raises 30% or more on the button, but only continues against a 3-bet with something like {AK, JJ+}. Here the opponent's tendency is folding too much to a 3-bet, and we can 3-bet profitably with any two cards because he folds so much.

    6. Toy Games

    Toy games in the context of poker are really simple versions of poker that usually have very strict rules on what can and cannot happen. The goal of analyzing toy games is to find insight for more complicated games. A toy game can also be modeled to represent a typical situation found in other poker games. In these cases, insight is sought out that applies in that particular spot. One example of a toy game used to analyze 3-betting situations preflop is to eliminate the option to call non-all-in bets.

    7. Half-Street Games

    A half-street game is a heads-up toy game that occurs over one street of betting that requires that the first player checks in the dark. An example of solving a simple half-street game for optimal strategy is found here http://www.spoonitnow.com/stuff/Example_11-1.rtf. Some introductory notes on these games and their analysis can be found at http://www.spoonitnow.com/stuff/Introductory_Notes.rtf.

    8. [0,1] Games

    A [0,1] game is a game in which the players' ranges are represented as a random real number on the interval from 0 to 1. A trivial [0,1] game is solved for optimal strategy here http://www.spoonitnow.com/stuff/Example_11-2.rtf.

    9. Game Theory Concerns for Micros

    The main thing game theory can help you do at micro stakes is to figure out the best way (or at least a winning way) to play against a certain range. While the actual act of deciding your opponent's range is more of an exercise in reasoning and a small amount of statistics, the most effective usage for that information once you get it has its basis in game theory. To that end, an immediate way you can put game theory at work for you is to set up sample ranges in different spots and figure out the best way to play against those ranges. Most of the time when you're analyzing a hand, this is what you should be doing anyway.


    So now that you've had some introduction to game theory, what sorts of toy games can you think of that can give us insight into no-limit hold'em?
  2. #2
    Ouch, spoon, it's too damn late at night to think up decent toy games. I'll try tomorrow. Nice post, btw.

    For anyone interested, rock-paper-scissors is also called roshambo (or rochambeau). An extremely interesting game in a similar vein is the Italian game called Morra. I like Morra better, and I think I can make it into a toy game, but not tonight.
  3. #3
    spoonitnow's Avatar
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    Here's one that I alluded to in the OP.

    Suppose with 100bb stacks it folds to Villain on the button who opens to 4x. Hero 3-bets to 12x in the SB, and the BB folds. Villain's options are to 4-bet to 32x or fold, and Hero's options if Villain 4-bets are to 5-bet to 100bb or fold. Finally, if Hero 5-bets then Villain has the choice between calling or folding. If Villain folds, the flop/turn/river is dealt and there is a showdown.

    Now, suppose Villain raises with any Ace, any two cards 9 or higher, and any pocket pair, 4-bets with TT+, AQ+, and calls a shove with his entire 4-betting range. How can Hero maximize his value?

    After you figure this out and start to get a feel for how adding different hands to different ranges will affect Hero's value, change Villain's ranges and do it again.
  4. #4
    settecba's Avatar
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    great post spoon...I had just finished reading the chapter about bluffing and game theory of Sklanskys TOP and you post this... I find this very interesting and I am actually going to take the class "Game Theory" next semester.
    Quote Originally Posted by ISF
    Getting good at poker is like that scene in the matrix where Neo suddenly sees that everyone is just a bunch of structured numbers and then he starts bending those numbers in really weird ways.
  5. #5
    settecba's Avatar
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    example 11-3 link is not working...
    Quote Originally Posted by ISF
    Getting good at poker is like that scene in the matrix where Neo suddenly sees that everyone is just a bunch of structured numbers and then he starts bending those numbers in really weird ways.
  6. #6
    spoonitnow's Avatar
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    Quote Originally Posted by settecba
    example 11-3 link is not working...
    Sorry about that. I checked and the file was corrupted or something and I've reformatted my computer since then so I don't have a backup copy, sorry.
  7. #7
    settecba's Avatar
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    Quote Originally Posted by spoonitnow
    Here's one that I alluded to in the OP.

    Suppose with 100bb stacks it folds to Villain on the button who opens to 4x. Hero 3-bets to 12x in the SB, and the BB folds. Villain's options are to 4-bet to 32x or fold, and Hero's options if Villain 4-bets are to 5-bet to 100bb or fold. Finally, if Hero 5-bets then Villain has the choice between calling or folding. If Villain folds, the flop/turn/river is dealt and there is a showdown.

    Now, suppose Villain raises with any Ace, any two cards 9 or higher, and any pocket pair, 4-bets with TT+, AQ+, and calls a shove with his entire 4-betting range. How can Hero maximize his value?

    After you figure this out and start to get a feel for how adding different hands to different ranges will affect Hero's value, change Villain's ranges and do it again.
    Where should I start to solve this? And also...does the question mean what is Heros 3betting and 5betting range in order to maximize his value?

    Ive been thinking about this, but i could use some help getting started.

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