Calling or Bluffing on the End

[tewall]

Here are some ideas which may be helpful for making a decision at the end (assuming no future bets possible) as to whether to bluff or call. I proofread this several times, but I might have made a mistake somewhere. It's a challenge to keep the different values straight. I'll be happy to address any points that need clarification (i.e., correct any goofs).

We assume the caller has a hand which will beat a bluff, and the bluffer's hand can only win if the opponent folds. A is the better, and B the caller. First we'll consider the case of calling.

Let b represent the bet amount required to call, or the amount of the bluff (they are the same of course). Let P1 represent the pot before A bets, P2 the pot after B bets, and P3 the pot after B calls.

B is being given odds of b to win P2. Since B can only win if A is bluffing, B should call if the odds that B is bluffing is greater than b/P3. If the odds are less than b/P3 that A is bluffing, B should fold. If the odds are exactly b/P3, then it makes no difference.

This ratio, b/P3, represents the optimum uninformed bluffing strategy for A. That is, if B is following the correct uninformed calling strategy (discussed later), then A's optimum bluffing strategy is to bluff b times for every P2 times a value bet is made. If A bluffs less than this, B profits by always calling, and conversely by always folding if A bluffs more often than this.

Now we'll consider bluffing. The bluffer is risking b to win P1. He can only win if B folds, so this risk is profitable if B will fold more often than b in P2 times. Or, in other words, if for every P2 bets, B folds b times and calls P1 times, A will break even. This is because the b times B folds, A wins P1, while the P1 times B calls, A loses b. bP1 = bP1 for both sides of the equation, so A breaks even. This break even point represents the correct uninformed calling strategy referred to earlier.

We're now ready to formulate calling and bluffing strategies.

If you are the caller:
1) Estimate the probability your opponent is bluffing.
2) If that estimate is less than b/P3, fold. Otherwise call.
3) If you are unable to make an estimate, call P1 times for every b times you fold.

If you are the bettor:
1) Estimate the probability your opponent will fold.
2) If that estimate is greater than b/P2, bluff.
3) If you are unable to make an estimate, bluff b times for every P2 times you value bet.

Here's some practical examples. There's $50 in the pot for each example.

Your opponent bets $50. Should you call? b=50 and P3=150, so if you estimate his chances of bluffing are less than 1 in 3 you should fold. If you can't make an estimate, call 1/2 of the time.

Your opponent bets $100. Should you call? b=100 and P3 = 250, so if you estimate his chances of bluffing are less than 2/5 you should fold. If you are unable to make an estimate, call 1/3 of the time.

Your opponent bets $25. Should you call? b=25 and P3 = 100, so if you estimate his chances of bluffing are less than 1/4 you should fold. If you are unable to make an estimate, call 2/3 of the time.

Should you bluff? If so, how much?

If you bet 50, you make a profit if your opponent will fold more than half the time. If you bet 100, you make a profit if your opponent will fold more than 2/3 of the time. If you bet 25, you make a profit if your opponent will fold more than 1/3 of the time. If you don't know what your opponent will do, your best uninformed strategy is that 1/3, 2/5, and 1/4 of your bets should be bluffs respectively.

There's a useful visual image which makes the calculations easy to do, but it's a bit hard to explain. I'll make an attempt. Let xxxxxx represent the pot and xxx the bluffers bet. The caller should call xxxxxx times for every xxx times he folds. The bluffer should bluff xxx times for every xxxxxx + xxx times he value bets. If either side's bluffing/calling pct. differs from this, the other side can profit.

The key thing to keep in mind is each player bases his optimum behavior on the relation of the other guy's bet/call to the pot. In other words, A's best uninformed bluffing strategy is the amount of B's call over the amount in the pot after B's call while B's best uninformed calling strategy is the amount in the pot before A's bet over the amount in the pot after A's bet.

[Paul2432]

Nice post. One question. It would seem the more often you can value bet on the river, the more often you can bluff. How often does a good player value bet the river? This number should give you a discount factor for the bluffing frequency.

For example, suppose a good player will value bet 1/3 of the time on the river. Then a pot sized bluff should be made 1/6 of the time you reach the river. So out of six times reaching the river, bluff once, value bet twice, and check three times.

Now that I think about it more though, the above isn't really right. You really need a different value betting frequency for every possible combination of betting/board cards.

This is pretty complicated. How do you use the bluffing frequency in practice?

Paul

P.S. I know in closed games like stud or draw, its much simpler, as you can use the cards to get the right ration of bluffs to value bets. In hold'em though a busted draw is obvious, so this won't work.

[tewall]

1/3 of the pot-sized bets should be bluffs, if you're following the best uninformed bluffing strategy. So if a good player would value bet 1/3 of the time on the river, he should bluff 1/9 of the time on the river.

As you figured out, you can't figure out how often you'll value bet on the river. It will change depending on the situation.

In practise, it really isn't very complicated, because you'll rarely need to resort to the best uninformed strategies. You'll be able to make an estimate. So if the pot is laying you 2 to 1 odds to make a call, you just ask yourself if you think it's more or less than 1/3 likely your opponent is bluffing, and make your decision.

If you have a close decision, you could fall back on the best uninformed calling strategy, but in practise you're usually better off trying to think through the hand and reason what your opponent is likely to hold, and that will sway your decision one way or the other.

The one time knowing the best uninformed strategies is really helpful is when you are totally outclassed by your opponent. If he is much better than you are, you can cut down a lot of his edge by using randomizing techniques.