Elizer, I accept your general point, but in your coin flipping example, it was unclear to me what your trade would be. Would you bet on heads or tails and in what circumstances? In a real world scenario I suspect its more likely to be the opposite of what I think you suggested but that is not entirely clear to me.
It all depends on where the market price is. There is a theory formed by a relatively respected speculator called “reflexivity,” basically that markets tend to perpetuate trends and overshoot fair values http://www.geocities.com/ecocorner/intelarea/gs1.html). And there is some other evidence of this in books such as Bob Schiller’s Irrational Exuberance about information cascades etc. So given there have been 9 heads in a row, maybe your average bear would think its more likely to come up heads than it’s genuine expected value, so I would argue that the market would probably overvalue the true likelihood (which according to you is 10/11ths), and so they would bet on heads and you would want to be short (bet on tails) if their expectation/price is greater than 10/11ths. The difference between their price and your price is called the edge, and you would capture that edge by betting on tails. So it would be a relatively modest profit opportunity. Not the other way round as I think you described, where you seem to be saying that the market would be pricing it at 50/50 and fair value is 10/11ths and you should bet on heads. It all depends on where the market would trade, above fair value or below fair value, I say above, in which scnenario one should bet on tails, and there is a minor profit opportunity, you seem to be saying below, bet on heads, and there is a major profit opportunity. A related question is how much should you actually bet?
Anyway, maybe just a case of my misunderstanding things. Another, perhaps, simpler way to think of markets are as beauty contests, where you are trying to pick the winner of the contest, which is not the girl you think is most beautiful, but the girl that everybody else thinks is the most beautiful: http://en.wikipedia.org/wiki/Keynesian_beauty_contest
So given there have been 9 heads in a row, maybe your average bear would think its more likely to come up heads than it’s genuine expected value, so I would argue that the market would probably overvalue the true likelihood (which according to you is 10/11ths), and so they would bet on heads and you would want to be short (bet on tails) if their expectation/price is greater than 10/11ths.
That was the point of the coin flip example. It was to point out that the market is not random even if it appears to be about as random as a coin flip. Information from the previous flip factors into the next flip, reducing the likelihood that any given trend will continue.
What I think you are missing is the fact that everybody knows that the way to take advantage of an inflated price is to sell short—it is not a unique insight on your part. Since it’s common knowledge, obviously everybody knows that everybody else knows that the way to exploit this situation is to sell short. Therefore there is a very high probability that a significant portion of the market will bet on tails to capture the edge. This destroys your likelihood of successfully beating the edge, because too many people are going to attempt the same thing you are. Those who recognize this (and there are many) also know that the next level of exploitation is to bet on heads.
The end result is a wash—there is a 50⁄50 chance that the 11′th flip will be heads, not a 10⁄11 chance like traditional probability suggests, because everybody is trying to out-exploit everybody else. The market is anti-inductive.
Think of poker. Someone who knows the probabilities of poker hands can win in far more situations than someone who does not. They will know when to bet and when to fold, maximizing their success. However, when playing with players who know the probabilities of poker hands this strategy becomes much less successful. Because everyone is only playing cards they can win with, it is only the really lucky players who get more good cards than bad who come out ahead.
However, by exploiting the likelihood of given cards, they can pretend they have cards they do not have and convince the rest of the players to give up. This makes their probability of winning skyrocket, and the net result is that the best hand winds far less often than probability suggests. This is the result of everybody knowing how to exploit the probabilities—everybody knows to go for the edge.
In high level poker, however, the best hand wins 99% of the time. Each particular hand wins at almost exactly the rate its likelihood of appearing suggests. Why? Because everybody at the table knows the probabilities for a given hand, and everybody is going to attempt to exploit those probabilities. Furthermore, everybody knows that everybody knows this, so it is only on extremely rare occasions that someone is actually able to exploit the probabilities and win with a weaker hand. In this scenario it is extremely difficult to inflate the value of the cards you are holding by bluffing, because the person you are bluffing knows the likelihood that you have the hand you are pretending to have and can compare that to their own hand to get their chances of winning. It still works on occasion though, and can be pretty spectacular.
This is the efficiency of the market. It is anti-inductive because information flows freely. As soon as an exploit is discovered it is nullified by the fact that it cannot be hidden, and everybody will therefore take advantage of it.
Mostly because its not actually true. If bluffing only worked 1% of the time, then no-one would bluff, so people would just fold mediocre hands against bets, so bluffing works again. If you solved some simplfied version of poker, so everyone is playing according to the exact Nash equibrium, there would still be plenty of bluffing.
That was four years ago, but I’m pretty sure I was using hyperbole. Pros don’t bluff often, and when they do they are only expecting to break even, but I doubt it’s as low as 2% (the bluff will fail half the time).
I’d also put in a caveat that the best hand wins among hands that make it all the way to the river. There are plenty of times where a horrible hand like a 6 2, which is an instant fold if you respect the skills of your fellow players, ends up hitting a straight by the river and being the best hand but obviously didn’t win. Certainly more often than 1%, and there are plenty of better hands that you still almost always fold pre-flop that are going to hit more often.
So, at best it was poorly stated (i.e. hyperbole without saying so), at worst it’s just wrong.
I know you’re using hyperbole, but I’m going to do the calculations anyway :)
If you bet a fraction x of the pot, with prob p of winning, and no outs, then your EV is p-(1-p)x. Clearly, EV>0 for optimal play, and a half-pot sized bet is common, so p-(1-p)/2>0 ⇒ p>1/3.
So the bluff should succeed at least 1⁄3 of the time.
Now suppose I have made some large bets, and you think I have at least JJ with 95% prob, and am bluffing with junk with 5% prob. I think you can beat JJ with 30% probability. I might chose to bet half the pot with all my possible hands (I’m now playing a probability distribution, not a hand), in which case you have to fold with 70% of your hands because 0.05 (1+0.5)<0.95 1. So in this case, my bluff succeeds 70% of the time, with EV 0.7-(1-0.7)/2=0.55.
Of course this is a massively simplified example.
Apparently, according to a book I read, if two pros playing head up no-limit are dealt 9 4, the author estimated that the person who has position (plays second) has around 2⁄3 chance of winning by bluffing his opponent off the hand, and of course the person who plays first might win by bluffing as well. So this seems to indicate that there is a reasonable chance to win by bluffing.
Overall, I think pros don’t make so many dramatic all-in bluffs, and in fact tend to semi-bluff, by betting with hands that have outs anyway.
Elizer, I accept your general point, but in your coin flipping example, it was unclear to me what your trade would be. Would you bet on heads or tails and in what circumstances? In a real world scenario I suspect its more likely to be the opposite of what I think you suggested but that is not entirely clear to me.
It all depends on where the market price is. There is a theory formed by a relatively respected speculator called “reflexivity,” basically that markets tend to perpetuate trends and overshoot fair values http://www.geocities.com/ecocorner/intelarea/gs1.html). And there is some other evidence of this in books such as Bob Schiller’s Irrational Exuberance about information cascades etc. So given there have been 9 heads in a row, maybe your average bear would think its more likely to come up heads than it’s genuine expected value, so I would argue that the market would probably overvalue the true likelihood (which according to you is 10/11ths), and so they would bet on heads and you would want to be short (bet on tails) if their expectation/price is greater than 10/11ths. The difference between their price and your price is called the edge, and you would capture that edge by betting on tails. So it would be a relatively modest profit opportunity. Not the other way round as I think you described, where you seem to be saying that the market would be pricing it at 50/50 and fair value is 10/11ths and you should bet on heads. It all depends on where the market would trade, above fair value or below fair value, I say above, in which scnenario one should bet on tails, and there is a minor profit opportunity, you seem to be saying below, bet on heads, and there is a major profit opportunity. A related question is how much should you actually bet?
Anyway, maybe just a case of my misunderstanding things. Another, perhaps, simpler way to think of markets are as beauty contests, where you are trying to pick the winner of the contest, which is not the girl you think is most beautiful, but the girl that everybody else thinks is the most beautiful: http://en.wikipedia.org/wiki/Keynesian_beauty_contest
That was the point of the coin flip example. It was to point out that the market is not random even if it appears to be about as random as a coin flip. Information from the previous flip factors into the next flip, reducing the likelihood that any given trend will continue.
What I think you are missing is the fact that everybody knows that the way to take advantage of an inflated price is to sell short—it is not a unique insight on your part. Since it’s common knowledge, obviously everybody knows that everybody else knows that the way to exploit this situation is to sell short. Therefore there is a very high probability that a significant portion of the market will bet on tails to capture the edge. This destroys your likelihood of successfully beating the edge, because too many people are going to attempt the same thing you are. Those who recognize this (and there are many) also know that the next level of exploitation is to bet on heads.
The end result is a wash—there is a 50⁄50 chance that the 11′th flip will be heads, not a 10⁄11 chance like traditional probability suggests, because everybody is trying to out-exploit everybody else. The market is anti-inductive.
Think of poker. Someone who knows the probabilities of poker hands can win in far more situations than someone who does not. They will know when to bet and when to fold, maximizing their success. However, when playing with players who know the probabilities of poker hands this strategy becomes much less successful. Because everyone is only playing cards they can win with, it is only the really lucky players who get more good cards than bad who come out ahead.
However, by exploiting the likelihood of given cards, they can pretend they have cards they do not have and convince the rest of the players to give up. This makes their probability of winning skyrocket, and the net result is that the best hand winds far less often than probability suggests. This is the result of everybody knowing how to exploit the probabilities—everybody knows to go for the edge.
In high level poker, however, the best hand wins 99% of the time. Each particular hand wins at almost exactly the rate its likelihood of appearing suggests. Why? Because everybody at the table knows the probabilities for a given hand, and everybody is going to attempt to exploit those probabilities. Furthermore, everybody knows that everybody knows this, so it is only on extremely rare occasions that someone is actually able to exploit the probabilities and win with a weaker hand. In this scenario it is extremely difficult to inflate the value of the cards you are holding by bluffing, because the person you are bluffing knows the likelihood that you have the hand you are pretending to have and can compare that to their own hand to get their chances of winning. It still works on occasion though, and can be pretty spectacular.
This is the efficiency of the market. It is anti-inductive because information flows freely. As soon as an exploit is discovered it is nullified by the fact that it cannot be hidden, and everybody will therefore take advantage of it.
Really? This seems surprisingly high.
Mostly because its not actually true. If bluffing only worked 1% of the time, then no-one would bluff, so people would just fold mediocre hands against bets, so bluffing works again. If you solved some simplfied version of poker, so everyone is playing according to the exact Nash equibrium, there would still be plenty of bluffing.
That was four years ago, but I’m pretty sure I was using hyperbole. Pros don’t bluff often, and when they do they are only expecting to break even, but I doubt it’s as low as 2% (the bluff will fail half the time).
I’d also put in a caveat that the best hand wins among hands that make it all the way to the river. There are plenty of times where a horrible hand like a 6 2, which is an instant fold if you respect the skills of your fellow players, ends up hitting a straight by the river and being the best hand but obviously didn’t win. Certainly more often than 1%, and there are plenty of better hands that you still almost always fold pre-flop that are going to hit more often.
So, at best it was poorly stated (i.e. hyperbole without saying so), at worst it’s just wrong.
I know you’re using hyperbole, but I’m going to do the calculations anyway :) If you bet a fraction x of the pot, with prob p of winning, and no outs, then your EV is p-(1-p)x. Clearly, EV>0 for optimal play, and a half-pot sized bet is common, so p-(1-p)/2>0 ⇒ p>1/3.
So the bluff should succeed at least 1⁄3 of the time.
Now suppose I have made some large bets, and you think I have at least JJ with 95% prob, and am bluffing with junk with 5% prob. I think you can beat JJ with 30% probability. I might chose to bet half the pot with all my possible hands (I’m now playing a probability distribution, not a hand), in which case you have to fold with 70% of your hands because 0.05 (1+0.5)<0.95 1. So in this case, my bluff succeeds 70% of the time, with EV 0.7-(1-0.7)/2=0.55.
Of course this is a massively simplified example.
Apparently, according to a book I read, if two pros playing head up no-limit are dealt 9 4, the author estimated that the person who has position (plays second) has around 2⁄3 chance of winning by bluffing his opponent off the hand, and of course the person who plays first might win by bluffing as well. So this seems to indicate that there is a reasonable chance to win by bluffing.
Overall, I think pros don’t make so many dramatic all-in bluffs, and in fact tend to semi-bluff, by betting with hands that have outs anyway.