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Poker Strategy
Poker Strategy
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Guest
Guest
Apr 23, 2025
10:29 AM
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Since A knows B's hand, to optimize his expected value, he should bet always if he has a hand that beats B's hand. If A has a losing hand, the question is how he should play. Player A can choose to bet as a bluff or check. Considering only two options for A--to bet all the winning hands and to bluff always or to check always when having a losing hand--and only two options for B--to check/call always or to check/fold always--the payoff matrix which represents these pure strategies is as follows:
The payoffs for the other two strategy pairs are calculated similarly.
None of the strategy pairs is a Nash equilibrium:
The pair
(
((A, bet only winning hands) and (B, check/fold always)
)
) is not a Nash equilibrium since A can switch to the strategy (A, bet always) to increase his payoff if B keeps his strategy.
The pair
(
((A, bet only winning hands) and (B, check/call always)
)
) is not a Nash equilibrium since B can switch to the strategy (B, check/fold always) to increase his payoff if A keeps his strategy.
The reasoning behind the other two pairs of strategies is similar. To put it another way, player A's pure strategies can all be utilized. In order to protect himself from being exploited, player A should randomize his bluffing; that is, he should bluff with, say, chosen so that player B is indifferent between his pure strategies combined in a best-response mixed strategy, in which case B should have the same payoff with strategies (B, check/fold always) and (B, check/call always).
Similarly, if B employs pure strategies (B, always check/fold) or (B, check/call always), A can start bluffing all the time or bet only his winning hands to maximize the profit. B must employ a call-based mixed strategy to avert this scenario. The example shows how to apply game theory to find non-exploitative strategies. However, a balanced strategy does not necessarily result in maximizing the profit in practice--the opponent will not necessarily use the potential to counter-exploit an exploitative strategy.
Examples of Applications of Mathematics
In poker, conditional probability is frequently used to acquire information. Calculating probabilities to connect with a flop and complete draws, as well as learning about an opponent's range based on his previous actions, are typical examples. Proper application of conditional probability helps to implement our observations to reach the correct decision and can serve to devise a deceptive play based on the concept of levels of common knowledge.
Is your opponent bluffing?
You are faced with a decision to call a bet on the river. Based on your estimates, you are certain that your opponent has either missed his draw (the probability of which you estimate to be 70 %) or has a hand that beats yours (30%). Your hand beats a missed draw. Suppose you know that your opponent will always bet a winning hand and will bluff with a missed draw 20% of the time. Your adversary wagers. How can you estimate the likelihood that your opponent is bluffing if this is all you know? For practical purposes of the calculation, one can use a version of natural frequencies representation described in Bayesian theory in science and math. Check out best online poker.
Suppose you want to increase your chances of bluffing an observant opponent by faking a tell. You assume that your opponent has some basic knowledge of tells and considers nervousness as an indication of a strong hand. In order to influence his decision you pretend that your hands are shaking. Your opponent notices that your hands are shaking and modifies his estimate of the probability that you are bluffing based on new information.
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