Suppose I select particularly 6 key cards to bluff with. That means I will bet 24 times. That is, 18 times with the best hand and 6 times on a bluff. Thus, the odds against my bluffing are equally 3-to-1. The pot is $200 and when I bet there is $300 in the pot. Therefore your pot odds are also 3-to-1. You are calling $100 to win $300. Now when the odds against my bluffing are similar to the odds you are getting from the pot, it really doesn’t matter whether to call or fold. Though, whatever you do, you will even now lose exactly $600 after 42 hands. If you were to fold every time I bet, I would beat you out of $100 24 times when I bet and lose $100 to you 18 times, when I do not bet, for a net profit of $600. if you were to call me every time, you would beat me out of $200 six times when I’m bluffing and $100 18 times, when I do not bet, for a sum of $3000; but I would beat you out of $200 18 times when I bet with my fine hands for a sum of $3600. Once again I gain $600. So, other than being a psychic, nothing in the world can stop me winning that $600 per 42 hands and giving me a positive expectation of $14.29 per hand. Bluffing exactly 6 times out of 24 has changed a hand that was a 4-to-3 underdog when I never bluff at all into a 4-to-3 favorite – don’t care what approach you use against me.
Now, we will move into the core of the poker game theory and bluffing. Observe first that the percentage of bluffing I made was fixed – one time every 19 bets or 5 times every 23 bets or 7 times every 25 bets. Observe secondly that my bluffing was entirely random; it was based on some key cards I caught that my rival could not see. He would rather not know whether the card I drew was one of my 18 fine cards or a bluff card. Finally, observe what would have occur when I bluffed with clearly six cards – which made the odds against my bluffing in this particular case similar to the pot odds my rival was getting. In this specific instance my rival stood to lose exactly the same amount by calling or folding.
This is optimal bluffing poker strategy – it makes no difference how your rival plays. We can comprehend that if you come up with a bluffing strategy that makes your rival do exactly worse regardless of how he plays then you have an optimal strategy. And bluffing with an optimal strategy should be in such a manner that the odds against your bluffing are similar to the odds your rival is getting from the pot. In the case we have been discussing, I had 18 fine cards and when I bet my $100 producing a $300 pot, my rival was getting 3-to-1 odds from the pot. Thus, my optimal strategy was to bluff with six more cards, making the odds against my bluffing 3-to-1, similar to the pot odds my rival was getting.
Suppose the pot instead of $200 was $500 before I bet. Once again I had 18 winning cards, and my rival could only beat a bluff. The bet is $100, and therefore my rival would be getting $600-to-$100 pot odds when he called. At present, my optimal strategy would be to bluff with 3 cards. Having 18 fine cards and 3 bluffing cards, the odds against my bluffing would be 6-to-1, similar to the pot odds my rival would be getting to call when I bet. If I bet $100 and the pot were $100, I would have to bluff with 9 cards when I had 18 fine cards , making the odds against my bluffing similar to the 2-to-1 odds my rival would be getting from the pot.
It is essential to consider that when the effects are same whether your rival calls or folds, you will average the same regardless how that rival mixes up his calls and folds. Coming back to the example of optimal strategy, where I make a $100 bluff with 6 cards and bet 18 fine cards into a $200 pot, I will average $600 in profits per hands for a long time whether my rival calls 12 times and folds 12 times or calls 6 times and folds 18 times or whatever. The lack of ability of a poker players to find any response to counterbalance his disadvantage is the solution to the game theory problems, even though the other game theory books does not present in this form.
Bluffing on the basis of game theory can be expressed in terms of percentages. Suppose the pot is $100 and I bet $100 and your chance of making a hand is 25 percent. Therefore, if you bet, your rival is getting 2-to-1 odds from the pot. As there is a 25 percent chance of making a hand, there should be a 12 ½ percent chance you are bluffing to produce the 2-to-1 odds against your bluffing which is the optimal strategy of bluffing. For instance, in draw lowball there are 48 cards you do not see when you draw one card, and presume 12 of them (25 percent) will make your hand. Therefore, for a bluffing you should select 6 other cards (12 ½ percent) out of the 48 cards.
Obviously, you select cards to randomize your bet in poker. Without the random factor, the fine rivals whom you use game theory to bluff would immediately select your plan and destruct you. The essence of the game theory is that even if your rival knows you are using it, he can do nothing about it.
Game Theory and Bluffing Frequency as per Your Rivals
In real poker situations, optimal strategy based on the game theory is not always the best approach. Of course, if you are up against a rival who always calls you, then you should not bluff with him any longer. Similarly, if you are up against a rival who folds frequently, you should bluff with some frequency.
Game theory works out these shifts in strategy. Observe in the first section of this chapter that if you bluffed with five cards instead of six – that is a bit less than optimally – you will win $300 more per 42 hands if your rival calls rather than folds each time. On the other hand, instead of six if you bluffed with seven cards, you will win $300 more if your rival folds rather than calls each time. This is where a player’s decision supersedes optimal game theory strategy: He would bluff slight less against rival who call too much and a slight more against rivals who fold too much.
Good, shrewd players comprehend this concept. If they found they have folded on the end a few hands in a row, they are acceptable to call next time. Or else other players will start bluffing them. And they use same principle to decide whether to bluff them. It is against such professional players, whose calling and folding are correct on target, or whose decision is as good as or better than yours, that game theory becomes the correct tool. So when you see it, there is no means that they can outplay you.
Review of Game Theory as a Tool for Bluffing
By using game theory to decide whether to bluff, you must first determine your chances to make your hand. Afterward you must determine the odds your rival is getting on that bet. Then you must bluff in such a manner that the odds against your bluffing are similar to your rival’s pot odds.
Let’s take another example. Suppose you bet $25 with the $100 in the pot and you have a 20 percent chance of making your hand. Your rival is then getting $125-to-$25 or 5-to-1 odds if you bet. The percentage of your good hands to your bluff should be 5-to-1. As you have 20 percent chance of making your hand, you must bluff 4 percent of the time. (20 percent-to-4 percent comes to 5-to-1.) When you bluff in this manner, you certainly take huge advantage of the situation.
A good and safe method to randomize your bluffs, as seen before, is to select the cards from among those you haven’t seen. For instance, if ten cards make your hand and you need a 5-to-1 bluffing percentage, then you must select two more cards to bluff with.
Let’s take one more example. In draw poker, your rival has three cards and you draw one card to spade flush. Thus, the chances are more that your rival will not be able to beat a flush, only a bluff. There is $20 in the pot. The bet is $10. If you bet, your rival is getting $30-to-$10 or 9-to-3 from the pot. As nine unseen spades make your flush, you must select three more cards to bluff with, such as the two red 4s and the 4 of clubs. Now, you bet 12 cards producing a 9-to-3 proportion between your fine hands and your bluffs.
It is likely not possible to use cards to come at exactly the percentage you need to bluff optimally. However, the long you close, you can still hope to make a profit. Remember that my selecting six cards to bluff with in the draw lowball case produced exactly the correct ratio in relation to the pot odds my rival was getting; though, I still ended up with a gain when I bluffed with five or with seven cards whether my rival called or folded. Obviously, the nearer you are to the exact percentage, the better in terms of game theory.
Using Game Theory to Call Probable Bluffs
Not only you can use game theory to bluff but also you can use it to call probable bluffs. Generally, when your hand can beat only a bluff, you use your skill and judgment to decide the chances your rival is bluffing. If your hand beat some of your legitimate hands, then you do a normal difference of your chances of having the best hand in addition to the chances your rival is bluffing against the pot odds you are getting. However, against a rival whose decision is as good as or better than yours or one who is competent of using game theory to bluff, you can use game theory to prevent that player or at least reduce his profits.
For example, there is $100 in the pot, and your rival thinks you will fold one out of three times rather than call a $20 bet. It is profitable for that rival to come out bluffing $20 to win $100 because he tends to lose $20 twice but win $100 once for a net profit of $60 and an expectation of $20 per bet. Similarly, if he thinks you will never fold in this case, he will never bluff. Thus, it behooves you to have a rival think you might fold sometimes, but you must frequently call enough to chase his bluffs.
If you use a game theory to decide whether to call a probable bluff, you make estimation identical to those you make when determining whether to occupy a bluff yourself – and you randomize your call similar as you randomize your bluffs. You should make out what pot odds your rival is getting on his probable bluff, and you make the proportion of your calls to your folds exactly the same as the proportion of the pot to your rival’s bets. If your rival bets $20 to win $100, he is getting 5-to-1 on a bluff. So, you make the odds 5-to-1 against your folding. Which means you should call five times and fold once. You can take help of key games of cards to randomize again – for instance if you chase some unknown cards, you fold. Otherwise, you call.
On the contrary to use game theory to bluff, using game theory to determine whether to call cannot change a losing situation into a gainful situation. The game theory prevents your rival from outwitting you, similar in case as a coin in the odds-evens game prevents your rival from outwitting you there. But if the rival is using optimum game theory approach to bluff, there is nothing you can do to make the best of him.
A good decision cannot be replaced by the game theory. It can be used when you think your rival’s decision is as good as or better than yours or when you just do not know your rival. Moreover, game theory can also be used correctly to bluff or call a probable bluff only in a situation where the bettor certainly either has the best hand or is bluffing – for instance, in seven card stud a player betting into your pair of aces with a clear flush draw. However, if the bettor bets a legitimate hand which is not the best hand, then the concepts in Chapter Twenty-one, “Heads-Up on the End,” would apply.
When using game theory to decide whether to bluff, you must determine the pot odds your rival is getting if you bet and then randomly bluff in such a manner that the odds against your bluffing are similar to or almost similar to your rival’s pot odds. If you rival is getting 5-to-1, the odds against your bluffing should be 5-to-1. If you play poker in this fashion, you force your rival to take a wrong decision. He does just – or poorly – in the long run by calling or folding.
When using game theory to decide whether to call a probable bluff – suppose your hand can beat only a bluff and that your decision does not give the clue – you must determine the odds your rival is getting on a bluff. Make the proportion of your calls to your folds the same as those odds. If your rival is getting 4-to-1 pot odds on a bluff, you should randomly call four out of five times to make that bluffing not profitable.