Game Theory Applications to Poker
The concepts we have learned about game theory have been applicable in many situations, including war strategy, confessions to crimes, capital allocation in banks, etc. As a result, all of these categories can be considered “games.” In this blog post, I will analyze my favorite game, poker, through the lens of game theory.
Poker has been studied extensively from a mathematical standpoint since it is a game where ideas about strategy, decision theory, and probability are directly applicable. Poker strategies often involve acknowledging that there are a variety of possible hands the opponent could play, and in turn, our reaction to these hands. There exists game theory optimal strategies in two-person poker games, which we will now explore.
An example of GTO is described as follows:
In a game of poker, if Professor Easley has three of a kind and is wary of Professor Halpern having a flush, how often can Prof. Halpern bluff?
If Prof. Halpern has a 15% probability of having a flush and there are $500 in the pot, and Prof. Halpern can bet a fixed amount of $100, Prof. Easley can call when Prof. Halpern bets and potentially win $600. What if Prof. Halpern only bets when he has the flush? Prof. Easley could easily exploit this strategy by folding. If we define defensive value as the expectation of a strategy against the opponent’s most exploitative strategy, then Prof. Halpern has defensive value (15%)($500) = $75, and he only profits when he has the flush. If he bets on every hand instead, Prof. Easley has expected value (85%)(600) – $100 = $410 if he calls. If he folds, his expected value is 0, and so he will exploit Prof. Halpern’s strategy by calling every hand. Now his defensive value is (15%)(600) – $100= -$10. These are examples of pure strategies in poker.
In contrast, a mixed strategy in game theory is when the player does not have one definite action, but chooses based on the expected payoffs over their possible actions. Say that Prof. Halpern’s odds of bluffing are not fixed, which can be represented by p. The expected values for Prof. Easley’s calls when Prof. Halpern bets is calculated as follows:
E(Easley, call) = (p/(0.15+p)) * $600 – $100
The probability that Prof. Easley calls when Prof. Halpern bets is q. The expected values for Prof. Halpern can be calculated as follows:
q = (0.15 + p) * $600 – $100
Set p = q to obtain the optimal bluff frequency for Prof. Halpern.
What is the takeaway from this examination of poker through the lens of game theory? It is certainly not that the game theory concepts we learned in our second week of Networks is directly applicable to poker. To know the exact probability of having a flush, or any hand for that matter, is very difficult and virtually impossible to determine during a game of poker. We do learn, however, that in some games, there is a defensive value that may be more appropriate for the situation than the expected value. Also, play style is crucial to developing a mixed strategy in poker. Although calculating p and q seem infeasible during an actual poker game, real poker strategy revolves around the approximations of these figures, based on whether the opponent has either risk-seeking or risk-averse tendencies.
https://math.mit.edu/~apost/courses/18.204_2018/Jingyu_Li_paper.pdf