This is a real example that happened during Euro 2004. Now World Cup 2006 is coming up, so it strikes my mind again.
A friend of mine was betting on which team would win Euro 2004. He thought I'm interested in betting too (but he was wrong - I only enjoy watching soccer, but I hate soccer betting). One day he told me why he ended up putting his money on Germany.
"I think France will win. But France's odds are only 4:1 (meaning that the dealer would give you $4 for every $1 you bet, if France won), while Germany gives 15:1. So I'm betting on Germany."
At the first glance, this statement doesn't make sense at all. If you really think France will win, there is no way that you put your money on any team other than France. It doesn't matter even if Germany gives 1000:1 - if France wins, this is never realized.
So I am assuming that my friend has a subjective probability distribution, i.e. he doesn't know which team will win for sure, but his soccer knowledge enables him to assign winning probability to the teams. Suppose he is risk-neutral (meaning that he is indifferent between 1. $5 for sure and 2. a bet with 50% getting $10 and 50% getting nothing). In making his argument, he is likely to be thinking that France's chance of winning is lower than 1/4 and Germany's chance of winning is greater than 1/15. For example, if he thinks Germany has 10% (which is greater than 1/15) chance in winning Euro 2004, then he will benefit in the long run: suppose there are 100 of these events and each time he puts up $1. If he is right, approximately he will win 10 times, and the dealer is giving him 10*$15 = $150, which is greater than his $100 investment.
There is still a problem here. If the soccer betting market is efficient, the odds should reflect market expectation (this is the definition of "efficiency" in the finance literature); but part of my friend's expectation (France's chance of winning is lower than 1/4) is not reflected in the dealer's odds. (Note that, however, his expectation on Germany is counted, since the dealer will lower Germany's payout when he bets on it.) This is a potential explanation of why there's a huge illegal soccer betting market. Suppose my friend is a soccer expert and he is sure that France's chance of winning is only 20% (which is lower than the odds in the market, 1/4), he cannot make a profit by betting on France, but he can make profit by betting against France and making himself a dealer (which is illegal). If he accepts other people's bet at 1/4 and France wins only 1 out of every 5 times, then among 100 of these events, he collects $100 and pays only 20*$4 = $80.
To prevent illegal soccer betting, the most effective way is to make these activities not profitable. In fact, there are many betting websites that allow you to be on either side of the bet, i.e. you can be on the side that pays people when something happens, and you are effectively a dealer. (However, this is not legal in some sports and in some countries yet.) If everyone can be a dealer, he can bet whenever he doesn't agree with the market odds (My friend can bet against France in this case, and the market will be more efficient as his expectation is counted). There will be no economic justification of engaging in illegal betting markets.
April 27, 2006
April 26, 2006
Evolution
Are we still evolving? Depends.
For those of you who do not plan to have children, read my following argument.
The heart of evolution rests upon the notion, "survival of the fittest." Now that you have survived, and as long as you can provide your children with enough care, it is your obligation to reproduce; otherwise there is no meaning to evolution. The worst case is that the healthiest/smartest/wealthiest nowadays are among the most reluctant to have offspring. Imagine what would happen after several generations. Yes, the best genes survived, but they disappear once people die. Only inferior genes have a chance to get reproduced and pass along. Evolution works backwards. To say that we are going back from humans to apes would be an exaggeration, but it might be the case that one day we find our world filled with people who are less fit.
The extreme evolutionist might go on and argue that we should never cure inherited diseases, or at least should ask people infected with these diseases not to have children. This helps the evolutionary process by eliminating the "bad" genes. However, this becomes more like a moral issue. The Oath of Hippocrates, i.e. the "doctor's oath," clearly states that "into whatever houses I enter, I will go into them for the benefit of the sick." The doctor is obliged to cure all diseases whenever he is able to. Everyone has the right to live and the right to give birth to children. If they are deemed not to fit the world, natural forces will come into play. It is not you or me or the doctor who makes the decision.
Again, if you have the ability to raise a kid, it is not your decision to have one or not; it is your natural obligation since first signs of life appeared on Earth 3.6 billion years ago. Evolution is a continuous process: it should never be altered by human beings, the species that benefit from it the most.
For those of you who do not plan to have children, read my following argument.
The heart of evolution rests upon the notion, "survival of the fittest." Now that you have survived, and as long as you can provide your children with enough care, it is your obligation to reproduce; otherwise there is no meaning to evolution. The worst case is that the healthiest/smartest/wealthiest nowadays are among the most reluctant to have offspring. Imagine what would happen after several generations. Yes, the best genes survived, but they disappear once people die. Only inferior genes have a chance to get reproduced and pass along. Evolution works backwards. To say that we are going back from humans to apes would be an exaggeration, but it might be the case that one day we find our world filled with people who are less fit.
The extreme evolutionist might go on and argue that we should never cure inherited diseases, or at least should ask people infected with these diseases not to have children. This helps the evolutionary process by eliminating the "bad" genes. However, this becomes more like a moral issue. The Oath of Hippocrates, i.e. the "doctor's oath," clearly states that "into whatever houses I enter, I will go into them for the benefit of the sick." The doctor is obliged to cure all diseases whenever he is able to. Everyone has the right to live and the right to give birth to children. If they are deemed not to fit the world, natural forces will come into play. It is not you or me or the doctor who makes the decision.
Again, if you have the ability to raise a kid, it is not your decision to have one or not; it is your natural obligation since first signs of life appeared on Earth 3.6 billion years ago. Evolution is a continuous process: it should never be altered by human beings, the species that benefit from it the most.
April 13, 2006
Time Travel
This has been bothering me for a while, mainly due to my obsession with the Japanese cartoon Doraemon, in which the main character Nobita has a time machine in the drawer of his desk.
Time travel is possible if one can travel faster than the speed of light. Suppose it is achieved one day in the future. Why don't we see future time travelers in our current world?
There is a discussion by Stephen Hawking in his famous book, A Brief History of Time. I didn't study enough Physics to comment on Hawking's suggestion, but let me state what my worries are.
If future time travelers can indeed go back in time, here's a couple potential explanations of why we don't see any of them now.
1. Our current world is not interesting enough to warrant them a visit.
2. They are here, but
a) They cannot reveal their identity; or
b) They are invisible to us.
If 1 is true, then we will probably see some of these travelers in the future, which I think is more intimidating than seeing aliens. 2a is possible since they might not be able to act upon their free will, as the past is fixed and they can only be quiet spectators. But the question now is: who are they? Scientists? Homeless people on the streets? Or you and me?
2b offers an explanation to why we don't identify them, but this is even more unpleasant. If this is true, you and I and everyone else have the chance of being watched by these future travelers, and we can't even tell if they are here. However, this poses more questions as well: How do they survive in our world? What makes them invisible to us?
Given the large amount of debate generated when cloning was invented, time travel will certainly be controversial. I hope this technique will never exist (or at least time travel is only confined to going to the future). Even God cannot go back in time, so shouldn't people.
April 11, 2006
Flaws in English
Two examples can show why English language might not be a very good one, at least in terms of logic.
Example 1 (Transitivity)
Logic says:
Premise 1 A>=B
Premise 2 B>=C
Conclusion A>=C
where >= can be interpreted as a Math operator (A is greater than or equal to C), or the preference relation in Economics (A is preferred to C), or anything that has this property, transitivity.
Let's see what happens if we apply this to English:
Premise 1 Having a house is better than having a car.
Premise 2 Having a car is better than having a dog.
Conclusion Having a house is better than having a dog.
Good! But...
Premise 1 Winning $1 is better than nothing.
Premise 2 Nothing is better than winning $100000.
Conclusion Winning $1 is better than winning $100000.
Premises 1 and 2 are correct in English, but the conclusion is obviously not true.
Example 2 (Implication)
Logic says:
A => B is true when
1. A is true and B is true, or
2. A is false and B is true, or
3. A is false and B is false.
where => is the logical operator "imply" (A implies B). In English it is translated to "If A, then B," e.g. "If he is your dad, then he is a male"; "If 18 is divisible by 6, then 18 is divisible by 3"; "If it rains, then I will bring an umbrella."
So far so good. Let's see how it can be problematic:
All of the following are true logically (reason stated in parentheses):
1. If I have $1 billion now, then I will give you $500 million. (Because I don't have $1 billion now)
2. If I have a dog now, then my dog has 5 legs. (Because I don't have a dog now)
3. If I am a male, then Bill Gates is a billionaire. (Because both parts are true, although they are not related)
(Bertrand Russell once publicly stated that if 1=2, then he was God. This is in the same spirit as 1.)
They all make no sense in English. Apparently, the translation of => into "if then" is not a good one. (Actually "imply" isn't a good name for it either.) There is no place in English for this fundamental logic operator.
Example 1 (Transitivity)
Logic says:
Premise 1 A>=B
Premise 2 B>=C
Conclusion A>=C
where >= can be interpreted as a Math operator (A is greater than or equal to C), or the preference relation in Economics (A is preferred to C), or anything that has this property, transitivity.
Let's see what happens if we apply this to English:
Premise 1 Having a house is better than having a car.
Premise 2 Having a car is better than having a dog.
Conclusion Having a house is better than having a dog.
Good! But...
Premise 1 Winning $1 is better than nothing.
Premise 2 Nothing is better than winning $100000.
Conclusion Winning $1 is better than winning $100000.
Premises 1 and 2 are correct in English, but the conclusion is obviously not true.
Example 2 (Implication)
Logic says:
A => B is true when
1. A is true and B is true, or
2. A is false and B is true, or
3. A is false and B is false.
where => is the logical operator "imply" (A implies B). In English it is translated to "If A, then B," e.g. "If he is your dad, then he is a male"; "If 18 is divisible by 6, then 18 is divisible by 3"; "If it rains, then I will bring an umbrella."
So far so good. Let's see how it can be problematic:
All of the following are true logically (reason stated in parentheses):
1. If I have $1 billion now, then I will give you $500 million. (Because I don't have $1 billion now)
2. If I have a dog now, then my dog has 5 legs. (Because I don't have a dog now)
3. If I am a male, then Bill Gates is a billionaire. (Because both parts are true, although they are not related)
(Bertrand Russell once publicly stated that if 1=2, then he was God. This is in the same spirit as 1.)
They all make no sense in English. Apparently, the translation of => into "if then" is not a good one. (Actually "imply" isn't a good name for it either.) There is no place in English for this fundamental logic operator.
April 10, 2006
Perhaps...
A Chinese version of this article was posted on my personal blog some two years ago. I like it so much that I have decided to translate it into English and use it as the first post on this blog. You can tell that this blog is going to be academic (or nerdy, since I'm in graduate school), not like the random ramblings on my other blog. If you are not interested in Finance/Econ/Math/Physics/Logic/Philosophy/History, stop reading and do NOT come back again. However, don't be scared/happy, I won't post research ideas or journal papers on here. It'll be more like a scientific discussion of anything that interests me.
Probability is my favorite topic. Let me start off by giving you a typical probability problem.
Problem There are 10 people in the queue. Two of them are named A and B. Suppose all possible queues are equally likely, e.g. the queues XXXAXXXBXX and XBXXAXXXXX carry the same probability (where X denotes the other 8 people). Also suppose that each person's position is independent of everyone else's (except that they cannot be in the same position in a particular queue), and everyone is equally likely to end up in a certain position (this is called the independently and identically distributed, or iid, assumption). What is the probability of having A lining up in front of B in this queue?
Solution Combinatorics approach: Count all the favorable events, e.g. A ranked 1st, B ranked 2nd, 3rd, ..., 10th; A ranked 2nd, B ranked 3rd, 4th, ..., 10th; etc., and then divide this by the number of possible queues, 10! (This is ten factorial, i.e. 10 x 9 x 8 x ... x 1, not TEN). This works nicely, doesn't it? But what if we have 100 people instead? 100! is astronomically large.
Alternative Solution A fifth-grader might have guessed the answer correctly. The probability is simply 0.5. Note that the event "A in front of B" and the event "B in front of A" are equally likely (they are iid), and the two events make up the entire probability space (it is either A in front of B or B in front of A). Probability axioms tell us that the probability has to be 0.5.
(Side note: If there is no reason to believe that one outcome is more likely than the other, then they must be treated as equally likely outcomes. This is the Laplace's Principle of Insufficient Reason, which tells us the probability of getting heads on coin tosses is 0.5 too.)
So, what does 0.5 mean? The most appropriate answer is: If you have a large number of these queues, say 10000, law of large numbers says approximately half of them (5000) will have A in front of B.
What if I only have one queue? How do I know if I see A first or B first? The most appropriate answer is: Who knows? The sample size (one) is too small for applying the law of large numbers.
Also note that the queues themselves are independent, which is the usual and logical assumption. That is, even if we have 10000 of these queues, the arrangements of the first 9999 queues have nothing to do with the 10000th queue. You never know how a particular queue behaves, despite that you have a large sample.
In real life, how often do we have a large sample (10000 queues) and only want to be approximately correct on the macro-level (knowing that about 5000 of them have A in front of B)? Under most circumstances, we are interested to know if something will occur or not (A in front of B in this queue). Sorry, probability is not able to handle this at all. Everywhere we can find examples of highly probable events not occurring, and seemingly improbable events happening. No one knows who decides what will/will not happen.
What makes things worse is the Heisenburg Uncertainty Principle. It states that you cannot simultaneously measure the momentum and the position of an electron. Roughly speaking, you cannot be 100% certain about something by just observing/measuring other things.
So what should we do when everything in the world is uncertain and probability doesn't help much? Nothing. That's why many people resort to God/supernatural forces/fate. Maybe they are right...
To make life easier, instead of calculating probability all the time, why not just "let it be"; instead of calling something "very likely," why not just say "perhaps." Isn't the world so much simpler and nicer?
Probability is my favorite topic. Let me start off by giving you a typical probability problem.
Problem There are 10 people in the queue. Two of them are named A and B. Suppose all possible queues are equally likely, e.g. the queues XXXAXXXBXX and XBXXAXXXXX carry the same probability (where X denotes the other 8 people). Also suppose that each person's position is independent of everyone else's (except that they cannot be in the same position in a particular queue), and everyone is equally likely to end up in a certain position (this is called the independently and identically distributed, or iid, assumption). What is the probability of having A lining up in front of B in this queue?
Solution Combinatorics approach: Count all the favorable events, e.g. A ranked 1st, B ranked 2nd, 3rd, ..., 10th; A ranked 2nd, B ranked 3rd, 4th, ..., 10th; etc., and then divide this by the number of possible queues, 10! (This is ten factorial, i.e. 10 x 9 x 8 x ... x 1, not TEN). This works nicely, doesn't it? But what if we have 100 people instead? 100! is astronomically large.
Alternative Solution A fifth-grader might have guessed the answer correctly. The probability is simply 0.5. Note that the event "A in front of B" and the event "B in front of A" are equally likely (they are iid), and the two events make up the entire probability space (it is either A in front of B or B in front of A). Probability axioms tell us that the probability has to be 0.5.
(Side note: If there is no reason to believe that one outcome is more likely than the other, then they must be treated as equally likely outcomes. This is the Laplace's Principle of Insufficient Reason, which tells us the probability of getting heads on coin tosses is 0.5 too.)
So, what does 0.5 mean? The most appropriate answer is: If you have a large number of these queues, say 10000, law of large numbers says approximately half of them (5000) will have A in front of B.
What if I only have one queue? How do I know if I see A first or B first? The most appropriate answer is: Who knows? The sample size (one) is too small for applying the law of large numbers.
Also note that the queues themselves are independent, which is the usual and logical assumption. That is, even if we have 10000 of these queues, the arrangements of the first 9999 queues have nothing to do with the 10000th queue. You never know how a particular queue behaves, despite that you have a large sample.
In real life, how often do we have a large sample (10000 queues) and only want to be approximately correct on the macro-level (knowing that about 5000 of them have A in front of B)? Under most circumstances, we are interested to know if something will occur or not (A in front of B in this queue). Sorry, probability is not able to handle this at all. Everywhere we can find examples of highly probable events not occurring, and seemingly improbable events happening. No one knows who decides what will/will not happen.
What makes things worse is the Heisenburg Uncertainty Principle. It states that you cannot simultaneously measure the momentum and the position of an electron. Roughly speaking, you cannot be 100% certain about something by just observing/measuring other things.
So what should we do when everything in the world is uncertain and probability doesn't help much? Nothing. That's why many people resort to God/supernatural forces/fate. Maybe they are right...
To make life easier, instead of calculating probability all the time, why not just "let it be"; instead of calling something "very likely," why not just say "perhaps." Isn't the world so much simpler and nicer?
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