Another way to see the bluff is that it reflects the principle: you have to pay to see a strong hand. Applying this principle, facing the raise of Bob, Alice must raise to pay Bob’s Full – and therefore, should also bluff when she decides to denounce the bluff of Bob, so as to force him to call and take advantage in average of his full.
The exchange feature is:
- (Alice) Opening the pot (I have at least a high pair) – the pot is 2.
- (Bob) Call (at least I have that) – the pot is 3.
- (Alice) Two cards (it is a pair or three of a kind).
- (Bob) A card (it’s a draw or two pair).
- (Alice) Opening the pot, so three (I have at least two high pairs) – the pot is 6.
- (Bob) I raise twice, so three more new (I declare to have succeeded my draw, but I’m bluffing once in three …) – the pot is 6 * 3 = 18, and Alice must add 9 to call.
- (Alice) I also raise twice the pot (36), so 9 plus 25 (I have a full house, but I‘m also bluffing once in three …) – the pot is (18) * 3 = 54, and Bob should add 25 to call.
At this point, Bob is always exposed to three basic choices: fold, call, or raise?
Except in special cases, a player who raised (to show a strong hand) has no objective reason to re-raise. If he does, it is for reasons that relate to the psychological play.
Fold or call (Alice rationally plays a full)
Alice raising twice the pot, the neutral point is to call one in three, regardless of the hand of Bob. In fact, if Alice plays rationally and made a raise, she is bluffing once in three, and a game winner the rest of the time. Calling a winning hand 66% is statistically expensive, but that’s the price to pay, because Alice is bluffing correctly, and that the bluff of Alice remains limited.
That said, Bob showed successfully completing his draw and as he bluffs rationally, the draw is two out of three. As to fold, do it only on hands that were losing anyway. Alice can bluff with a three of a kind, and it would be silly to call with a simple pair when you can make it in average with a straight … If there is a re-raise from Alice, Bob strategy is:
- If the initial raise of Bob was a bluff, fold.
- If the initial raise of Bob was justified by a flush or straight, call half the time the alleged full of Alice.
Raise (shows a four of a kind or a straight flush)
A rational raise shows a hand much stronger than that already showed by the opponent.
Regardless of the bluff, the first raise of Bob shows a successful draw – as the most likely form, that is to say, straight or flush. The revival of Alice shows her pair has improved by full or four of a kind, beating the straight or flush. An additional raise of Bob says he has a harder hand, which can only be a successful draw to the straight flush.
When asked “a card”, the four of a kind (served) or straight flush only occur in 0.25% of distributions: during the hundred shots that represents a party, improving a draw will see ten times, but a served four of a kind or an improvement to the straight flush is a hand we see only once in four parties. And it has to be just when Alice says she improves his hand to full? The coincidence is extraordinary …
The reasoning can be no more statistical, but must be psychological, because such a game will certainly not present twice at the table.
- Bob may have shown enjoy riding the bluff, for the pleasure of psychic game. It is enough to credit him with a probability of 0.5% to yield to the psychic in such a case for his bluff statistically permitted margin is exceeded, and that his bluff statistically justified to be sanctioned. If Alice actually holds a full house or a four of a kind, she can call to see the supposed straight flush of Bob on virtually any odd, of more a hundred against one.
- Bob may have shown a flawless game, worthy of a computer. Knowing that no system will allow him to attempt a bluff to four of a kind except at a credible rate of 0.12%, he prefers not to try at all this kind of bluff, too random. If the straight flush or four of a kind arise, he will enjoy a cold sweat of his opponents against its raises, but as he will never play this scene twice before the same audience he abandoned the idea of preparing a valuation of this type of hand by the sacrifice of pre bluffs. For Alice, suddenly, this game is rare enough that she chooses to limit the damage by never calling faced a rational adversary; if the choice is the wrong one, she will never know, because she will not have the opportunity to remake a similar one during the party.
- The raise of Bob may simply show – for one reason or another – that he does not believe the full of Alice. Bob may have felt too much hesitation of Alice on her raise, leading to think that this raise is itself a bluff. If that’s the case, Alice may decide to fold, which costs her nothing (since her raise, not being justified by a strong hand, was in any way an investment sacrificed): she shows his bluff with frequency statistically necessary to monetize draw of a full house another time, she shows that when she is bluffing it shows … mission accomplished, it is enough now to reproduce the same hesitations when she will have full house. She may also decide to raise Bob who will crack the first: if tha last has not simply called, it is because he too is bluffing.