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Mathematical expectation gain at poker

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This article details the mathematical relationships between leveling and expected gain in the case of poker.

Mathematical expectation

Law of large numbers

Poker is a game involving chance, and where financial gains depend on the skill at the game, the ability to read the psychology of the opponent, but also the probability of meeting favorable or unfavorable situations. The poker player must make decisions without certainty, based on only probabilities: The probability p(A1) that the situation is A1, p(A2) that the situation is A2, … And for each of these situations the possible gain is variable: a gain g1 for the position A1, a gain (or a negative loss) g2 for the situation A2

If the player has to make a decision, this decision will affect its future earnings. To compare the effect of these decisions, it usually consider the expected gain associated with the situation, which is mathematically defined as the sum of all earnings, weighted by the probabilities of corresponding situations:

E = Σip(Ai) · gi

The expected gain is the average of the gains that can be expected in similar situations: if you count the average of what will be won or lost on all comparable situations, this value will be found as the limit.

When the consequences of both choices have to be compared, the first comparison element is to calculate the expected gain resulting from one or other choice. Indeed, one can mathematically prove the following result: “A player who consistently choose the option for which the expected gain is necessarily maximum, will gain necessarily, after an infinite time, and provided you have an infinite capital.” The two conditions of the theorem (mathematically proven and unquestionable) look like a mathematician joke: no one has an infinite capital and forever to play. The practical result of this theorem, technically more difficult to demonstrate, is that:

A player who consistently choose the option for which the expected gain is maximum:

  • will be earning on average;
  • will be more clearly winning when will play a longer time;
  • he can play longer time if he will have a significant initial capital.

The practical consequence of these restrictions is that when a player is too tight financially speaking, the expectation of the game is no longer the only element to take into account: it must also (perhaps especially) take into account the probability of staying in the game, and not to be expelled from the game for lack of funds. Those who no longer have time or money have to use extreme tactics that do not enter the scope of this discussion. This restriction being made, we can consider that:

The optimum game is usually to play the game for which the expected gain is maximum.

Psychological and mathematical game

You can play poker against a computer, which every time makes the decision that maximizes his mathematical gain. The computer can be programmed to perfectly play based on factual information (bets, raises, numbers requested card …), but it does not incorporate subjective information (hesitations – sweating – through manholes – tremors in the voice – …) that would allow it to assume that this or that player is bluffing, or rather has a winning hand, independently of any consideration to statistical probabilities.

Against such a computer, one can imagine a player. Faced with poker expert computer, the player does not have to account for his behavior (nose scratching, shifty eyes, …) but only his game strategy. The restrictions on the scope of the theorem are same as in the previous section (must have much time and money), but the important result is:

Apart from the psychological factors: Whoever does not play based the maximum expected gain, loses on average.

On the other hand, poker is a game of “zero sum”: if we disregard the amount taken by the party organizer, (…), gains and losses of the party players compensate, and players have the same chances in the game. For this reason, the mathematically optimal play is a null gain:

If both players play solely on the expected gain, and ignore any psychological factor, they will have a zero gain on average.

The conclusion is that if we want to win at poker:


  • We must know the strategy that maximizes the expected gain;
  • Admittedly his opponent’s errors that do not maximize their expected gain, and sanction yhem.


  • We must suggest the opponent to get away from the optimum, wrongly;
  • We must not give the opponent indications that would allow him to get away from the optimum based on reason.

These alternatives lead to two styles of play, psychological and mathematics.

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