Game Theory and Bluffing

One would think that game theory is a theory about games, but it is really a section of mathematics dealing with the decision-making process.

While it applies to games, as we will see, it also applies to such disciplines as economics, international relations, social science and military science. Significantly, game theory try to find mathematically the best approach against someone also using the best approach. You think against your rival is weaker than are and it can be in any game wherever - you should depend on your judgment rather than on game theory. However, against a rival you think is better than you or against a rival you do not know, game theory can sometimes be able to overcome the other's judgmental surface.

In this regard, to know how game theory works, we will consider the children's game of odds and evens. Each of the two players may put his one or two fingers. If the total is odd then the rival wins and if the total is even then the other player wins. Now, mathematically this is an even game. However, in long series, it is possible for one person to achieve an surface by outwitting the other, by deciding whether to put out one or two fingers with a base what the other persons put out in the previous round or rounds by selecting up patterns - in other words, by figuring out what his rival is thinking and then putting out one or two fingers in order to frustrate him.

In case someone challenges you to this game. If you are confident about his judgment and ability to outguess you, he will lay you $101 to $100 per play. Let's say your challenger too have the best of it in terms of judgment. However, by employing game theory, you can joyfully agree the proposition with the guarantee that you have the best of it. The thing for you to do is to flip a coin to decide whether to put out one or two fingers.

If the coin comes as heads, you put one finger; if it comes up tails, you put two fingers. What this thing has done? It has entirely destructed your rival's ability to outguess you. The chances to put out one or two fingers are 50-50. The chances of a coin to come as head are also 50-50. However, other than thinking about whether to put out one or two fingers, the coin will make the decision for you, and most essentially it is randomizing the decisions. Your rival may be able outguess you, but you are forcing him to outguess an inanimate image that is impossible. One may even try to guess whether roulette will take position on the red or on the black.

As your challenger has laid you $101 to $100, by making a use of a game theory, you have assured yourself of a 0.5 percent mathematical advantage (or a 50-cent positive expectation per bet). You have separated whatever advantage your rival might have had in out-thinking you and giving yourself an insuperable surface over the long run. Only if you thought you could out think your rival would do better off by using your judgment instead of a coin flip.

Bluff by using Game Theory

We are extremely interested in this chapter with how game theory is applied in the art of bluffing and calling probable bluff in poker theory. Regarding this purpose, we will take in notice of mixed approach, an approach in which you can make a positive play - particularly a bluff or a call of a possible bluff - a fix percentage of time but you present a random element in order that your rival do not know when you are making the play and when you are not making the play.

Remember from the last chapter, everything else being equal, the player who bluffs so much and the player who never bluffs are on the extremely disadvantage against a player who bluffs perfectly. To explain this point and by using game theory one can decide how to decide correctly when to bluff, we will introduce a proposition.

We are playing draw lowball without joker, and I give you a pat:

I take a:

You stand a pat and I must draw one card. If I chase a five, a six, a seven, an eight, or a nine, I beat you with better low than yours. If I chase any other card, you win. It means that out of 42 cards remaining in the deck, I have 18 winners (4 fives, 4 sixes, 4 sevens, 3 eights, 3 nines) and 24 losers that make me to a 24-to-18 or 4-to-3 underdog. We each ante $100 but after the draw - which you do not see - I can bet $100.

For example, I'm going to bet $100 every time. Precisely, you would call each time because you would win $200 the 24 times I'm bluffing and lose $200 the 18 times I have the best hand for a total profit of $1200. However, if I said I will never bluff; I will only bet when I have your 9, 8 low best. Then you will fold each time when I bet, and once again you will win 24 times (when I do not bet) and lose 18 times (when I do) for a net profit of $600 as you win or lose $100 in each of these hands. Therefore, with either of these variations of the proposition, you certainly have the best of it.

But, if I bluff sometimes, the case would not be the same. Suppose I just want to bluff when I caught king of spades. In other words, whenever I caught any of the 18 fine cards and also when I caught the kings of spades then I would bet. If I bluff rarely, your correct play would be to fold when I bet because the odds are 18-to-1 against my bluffing. However, observe how this improves my position. Bluffing when I catch the king of spades does not give me a profit but it permits me to win 19 times instead of 18 and lose only 23 times instead 24 times. That one bluff out of 19 times has started to close the gap between your status as a favorite and mine as an underdog. Observe that you have no way of knowing when I am bluffing as I am randomizing my bluffs by using a card, an object as inanimate as the coin in the odds-evens game, to make my bluffing decisions for me.

If bluffing with one card makes me less of an underdog than never bluffing at all, suppose I select two - say, the king of spades and the jack of spades. Again your proper play is to fold when I bet. But, now you win only 22 times when I do not bet and I win 20 times when I do. Suppose you have no way of knowing when I am bluffing and when I am not, I use just two key cards to bluff, including 18 fine cards, has reduced you from 24-to-18 favorite to a 22-to-20 favorite - that is, from a 4-to-3 favorite to a 11-to-10 favorite.

This bluffing look likes to have some possibilities. Suppose, I picked five key cards instead of two cards - the king of spades and all the four jacks. It means that I would be betting 23 times - 18 times with the best hand and five times on a bluff. Now, all at once your situation is bad with your pat 9, 8 because you have to recognize whether I am bluffing when I bet. I could even inform you clearly the approach I am using, but you would still have to lose your money.

What would happen? You know very well that there are 18 cards which will make me my hand and five other free card that I will bluff with. So, the odds would be 18-to-5 or 3.6-to-1 against my bluffing. With my $100 bet and $200 in the bet, the pot is $300. So you are getting 3-to-1 odds from the pot. You cannot profitably call a 3.6-to-1 shot when you win only 3-to-1 for your money. Lo and consider, by using five cards to bluff with, I win that pot from you 23 out of 42 times and you win it only 19 times. I make a gain $400. Therefore, my infrequent random of bluffing has swung a hand that is a 24-to-18 underdog into a 23-to-19 favorite.

To be certain there is no mathematical sleight of hand here; you have to work out what happens if you call every time I bet. You will win $200 from me the five times I am bluffing and $100 from me the 19 times I do not bet for a sum of $2900. However, you will lose $200 to me the 18 times I have the best hand for a sum of $3600. Your net loss when you call is $700 which is $300 more than you lose if you just fold when I bet.

Instead of five, had I picked seven cards to bluff with, the odds then would be 18-to-7 against my bluffing and as the pot odds you are getting are 3-to-1, you would be forced to call when I bet. However, you have to end up losing! Seven times when I am bluffing, you would win $100 from me for a total of $1400 and 17 times I do not bet at all, you would win $100 from me for a total of $1700. Your wins after 42 hands would take you to the total of $3100. But I would win $200 from you the 18 times I bet with my good cards for a total of $3600, giving me a net profit and you a net loss of $500 after 42 hands.

It should be noticed - once again to be precise there are no tricks to this mathematics - that you would lose even more money if you folded every time I bet with my 18 fine cards and seven bluffing cards. You would win $100 from me the 17 times I do not bet, whereas I would win $100 from you the 25 times I do bet. Now your net loss would be $800 instead of $500.