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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?
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