Optimal Bluffing Frequency: Proof

The following scenario is an abridged version of the scenario that David Sklansky gives in The Theory of Poker, chapter 19. He states that your bluffing frequency should be equal to your opponent’s pot odds but he never gives a proof. Here is my attempt at providing such proof. Much of the following analysis will be his or based off of what he says (with the exception of the proof).

“You are playing heads-up poker with 1 card left to come. You will be first to act; you can either bet or check. Your opponent can only call or fold. There are dollars in the pot already. For the purpose of this example, assume there are cards left in the deck that give you a better hand. Furthermore, if your hand improves at all, it is guaranteed to be the best hand. We lose otherwise. What should our betting strategy be? Also assume that we tell our opponent exactly what our strategy is (and we aren’t liars, so we will follow that strategy)”

First observation: We should bet if we improve our hand because we will definitely be winners. However, if we never bluff (and he knows we never bluff), our opponent will never call if we bet, which makes betting our value hands useless. So it makes sense that we should bluff some times. But when?

We will use what card we get to signal whether or not we will bluff (assuming that card is not one of the cards that make our hand). So now assume that there are cards that make our hand, cards that we will bluff on, and cards that we will just check on (if we check, we lose).

To recap our strategy is as follows: Bet (and win) if we receive any one of our cards, bluff on the cards, and check on cards.

What is the expected value for an opponent who always calls?

The first term represents the value our opponent gets when we bluff and get called, the second term represents when we value bet and get called, and the last represents when we check.

What’s the value for an opponent that always folds?

If they always fold when we bet, they only win when we check.

Now back to our strategy. , the cards that make our hand, is a given of the problem, so our  decision consists of deciding the value of . I will claim, and provide some intuition, that we want to set a such that our opponent receives the same expected value from calling or folding.

So boiling this down to the to format, we get that our ratio of value bets to bluffs is to . This is precisely the formula for determining your pot odds.

Now why is it optimal to bluff such that your opponent receives the same EV from calling and folding? Well if this is the case, then any call/fold strategy your opponent employs can make at most (which is the EV of calling or folding when we use our strategy). If he calls all the time, he gets . If he folds all the time, he gets . If he calls half of the time and folds the other half, he gets . There is nothing he can do to do better. So that is a good thing. This gives us a good intuition that this should be a good point for us to push him towards. The fact that there is no better strategy for you involves using some facts about 2 player, zero-sum game that are too complicated to go into here. So just trust me on it.

An important note: This analysis assumes our opponent is very good and will play optimally for himself (this is usually the assumption of game theory). If you notice that an opponent generally calls too much or too little, it is a good idea to deviate from this strategy a little to take advantage of that fact.

So how useful is this idea at the table? We made some pretty strong assumptions about the scenario in order to derive this result, so it is not directly applicable. However, if you have a good feel for the hand you are in and the players at the table, it might be possible to establish these conditions. At that point, you should consider using such a strategy.

It also comes with my standard poker theory caveat: It may not be directly useful but it provides you with another tool to use at the table, should you feel that it is appropriate to use it.

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