Optimal Bluffing Frequency
If a player bluffs too infrequently, observant opponents will recognize that the player is betting for value and will call with very strong hands or with drawing hands only when they are receiving favorable pot odds. If a player bluffs too frequently, observant opponents snap off his bluffs by calling or re-raising. Occasional bluffing disguises not just the hands a player is bluffing with, but also his legitimate hands that opponents may think he may be bluffing with. David Sklansky, in his book The Theory of Poker, states "Mathematically, the optimal bluffing strategy is to bluff in such a way that the chances against your bluffing are identical to the pot odds your opponent is getting."
Optimal bluffing also requires that the bluffs must be performed in such a manner that opponents cannot tell when a player is bluffing or not. To prevent bluffs from occurring in a predictable pattern, game theory suggests the use of a randomizing agent to determine whether to bluff. For example, a player might use the colors of his hidden cards, the second hand on his watch, or some other unpredictable mechanism to determine whether to bluff.
Example (Texas Hold'em)
Again, let us return to the examples in The Theory of Poker:
when I bet my $100, creating a $300 pot, my opponent was getting 3-to-1 odds from the pot. Therefore my optimum strategy was . . . the odds against my bluffing 3-to-1.
Since the dealer will always bet with (nut hands) in this situation, he should bluff with (his) "Weakest hands/bluffing range" 1/3 of the time in order to make the odds 3-to-1 against a bluff.
Ex: On the last betting round (river), Worm has been betting a "semi-bluff" drawing hand with: A♠ K♠ on the board:
10♠ 9♣ 2♠ 4♣ against Mike's A♣ 10♦ hand.
The river comes out:
2♣
The pot is currently 30 dollars, and Worm is contemplating a 30 dollar bluff on the river. If Worm does bluff in this situation, he is giving Mike 2-to-1 pot odds to call with his two pair (10's and 2's).
Important: In these hypothetical circumstances, Worm will have the nuts 50% of the time, and be on a busted draw 50% of the time. Worm will bet the nuts (100%) of the time, and bet with a bluffing hand (using mixed optimal strategies):
Where s is equal to the percentage of the pot that Worm is bluff betting with and x is equal to the percentage of busted draws Worm should be bluffing with to bluff optimally.
Pot = 30 dollars. Bluff bet = 30 dollars.
s = 30(pot) / 30(bluff bet) = 1.
Worm should be bluffing with his busted draws:
x = Where s = 1
Assuming 4 trials, Worm has the nuts 2 times, and has a busted draw 2 times. (EV = Expected Value)
| Worm bets with the nuts (100% of the time) | Worm bets with the nuts (100% of the time) | Worm bets with a busted draw (50% of the time) | Worm checks with a busted draw (50% of the time) |
|---|---|---|---|
| Worm's EV = 60 dollars | Worm's EV = 60 dollars | Worm's EV = 30 dollars (if Mike folds) and -30 dollars (if Mike calls) | Worm's EV = 0 dollars (since he will neither win the pot, nor lose 30 dollars on a bluff) |
| Mike's EV = -30 dollars (because he would not have won the original pot, but lost to Worm's value bet on the end) | Mike's EV = -30 dollars (because he would not have won the original pot, but lost to Worm's value bet on the end) | Mike's EV = 60 dollars (if he calls, he'll win the whole pot, which includes Worm's 30 dollar bluff) and 0 dollars (if Mike folds, he can't win the money in the pot) | Mike's EV = 30 dollars (assuming Mike checks behind with the winning hand, he will win the 30 dollar pot) |
Under the circumstances of this example: Worm will bet his nut hand 2 times, for every one time he bluffs against Mike's hand (assuming Mike's hand would lose to the nuts and beat a bluff). This means that (if he called all three bets) Mike would win 1 time, and lose two times, and would break even against 2-to-1 pot odds. This also means that Worm's odds against bluffing is also 2-to-1 (since he will value bet twice, and bluff once).
Say in this example, Worm decides to use the second hand of his watch to determine when to bluff (50% of the time). If the second hand of the watch is between 1 and 30 seconds, Worm will check his hand down (not bluff). If the second hand of the watch is between 31 and 60 seconds, Worm will bluff his hand. Worm looks down at his watch, and the second hand is at 45 seconds, so Worm decides to bluff. Mike folds his two pair saying, "the way you've been betting your hand, I don't think my two pair on the board will hold up against your hand." Worm takes the pot by using optimal bluffing frequencies.
Please note: This example is meant to illustrate how optimal bluffing frequencies work. Because it was an example, we assumed that Worm had the nuts 50% of the time, and a busted draw 50% of the time. In real game situations, this is not usually the case.
The purpose of optimal bluffing frequencies is to make the opponent (mathematically) indifferent between calling and folding. Optimal bluffing frequencies are based upon game theory and the Nash Equilibrium, and assist the player using these strategies to become unexploitable. By bluffing in optimal frequencies, you will typically end up breaking even on your bluffs (in other words, optimal bluffing frequencies are not meant to generate positive expected value from the bluffs alone). Rather, optimal bluffing frequencies allow you to gain more value from your "value bets," because your opponent is indifferent between calling or folding when you bet (regardless to whether it's a value bet or a bluff bet).
Read more about this topic: Bluff (poker)
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