For a perfectly rational theoretical player (without bluffing, but very naive), the level of the bet and raise reflects directly the value of the hand. Indeed, the player feels a certain probability α when he holds the strongest hand at the table, given the auction and past bets. If the pot contains a sum P, he decided to bet an amount S based on the fact that his hand is stronger, strong enough for everybody “fold” after him – which will allow it to actually win the pot.
The highest amount that he can rationally bet is that on average he earn almost nothing, because on average the amount of P that he will win with a probability α is exactly the amount S that will be lost with probability (1-α) by a weak hand. In that case:
α.P = S(1 – α) and therefore Smax = P.(α/(1 – α)) or conversely αmax = S/(P + S)
A rational player (without bluffing) does not exceed this level of development. If the player bets more, it runs the risk that an opponent has a sufficient strength to play to “see” with the same bet, which will lose more than necessary. If it is rational, but very naive, it can play exactly this leveling. In this case, opponents can immediately deduce the probability α to which he considers his hand, and therefore whether or not revive. At minimum this level is the force that he wants to see his play.
In other words :
The level of bet of a rational supposed player allows to estimate the minimum level of his play: The ratio of a player’s bet and the pot level directly reflects the probability of minimal gain that that player believes for his play and thus his force he wants to display for his play.
This link between bid level and probability of winning is the main constraint of the game, virtually the only one. If a player bets too high compared to the value of his hand, and his opponents think they have the stregth to call “to see”, he will lose on average.
A player stronger than that level is interested to bet less than the value of his hand, for two reasons. First, to not discourage his opponents to play and restart, increasing on average the value of the pot. On the other hand, whenn he win, it does so at a reduced rate: the difference between his maximum raise and his real raise represents an average financial gain that should maximize, betting as low as possible – but without re-raising.
In other words:
The level of financially optimal raise of a strong game is the lowest bet that will allow the re-raising.
Re-raises and bluffs
Basically, a player raises when he thinks his hand is better than others, based on the play of the opponents. A raise is a priori economically viable if the probability of actually having the strongest game and win the stack justifies the level of raise.
That said, if the opponents do not call a raise it is not economically attractive compared to a simple call. Worse, if they will have a better hand, raising is a waste of money.
In fact, the rationality of a stimulus can be analyzed in relation to the ability to bluff: for a raise or call, there must be a bit of bluff in his game. In summary.:
- The interest of optimal bluffing strategy is to leverage statistically the winning hands, forcing the opponent to call to see them more often.
- A typical raising level is twice the value of the pot.
- The neutral point, a raise twice the pot is a bluff once out of three.
- The neutral point, a raise twice the pot must is called by one out of three.
A limit raise is one which on average is neither winning nor losing. A slightly stronger hand warrant to enter, a slightly weaker hand justifies opening raise at the lower (typically, half-pot).