Some days ago I was enjoying a coffee break with colleagues, when suddenly one of them made a comment about St. Petersburg paradox, as an example of the laws of expectations, which can throw out very strange results. So I decided to write something about it.
First, let's write the meaning of the word "paradox". Wikipedia says that Paradox is "a statement that apparently contradicts itself and yet might be true". This may sound confusing but maybe an example can help! What about the St. Petersburg paradox? Actually, this is also known as St. Petersburg lottery, and is as follows:
Imagine the typical game of tossing of coins. In this game, the coin is tossed until tails appears. Assume that our winnings increase, the longer the run of heads. So we win $1 if tails appears the first time, $2 if it come on the second, $4 on the third, and so on. The reward doubling each time. Also assume that you have to pay each time you play. The question is: what is a fair price for entry to the game?
The usual approach to solve it would be to calculate the game's expected reward. If that exceeds the entry price, then rationally you should agree to play, otherwise you shouldn't. This sounds reasonable, until we actually try to perform the calculation, when something very strange happens.
Each possible outcome has its own probability and reward: the probability that the game will end after one throw is 1/2, and this case, we would win $1. There is a probability of 1/4 that it will end after two throws, netting us $2. The probability is 1/8 after three throws, and we would get $4, and so on. Putting all these together, the expectation seems to come out as
(1/2 x $1) + (1/4 x $2) + (1/8 x $4) + (1/16 x $8) + ...
That is to say: $0.5 + $0.5 + $0.5 + $0.5 + ... In other words, the "expected" reward for this game is actually infinite! That's why this is called a "paradox". How can a simple game be infinitely profitable?Yet there is a sense in which it is correct. Again, it comes from repeated playing. Surprisingly, if you play the game often enough, then any entry price will indeed eventually represent good value. The catch is the number of rounds of the game needed. If the cost is $10 per game, you are likely to have to play over a million times before coming out in profit. If the cost is $100 per game, the lifetime of the universe is unlikely to be long enough. More about this and other paradoxes can be found in the book "Chaotic Fishponds and Mirror Universes" by Richard Elwes.
The Bernoulli family was involved in this problem. In fact, the problem was invented by Nicolas Bernoulli in 1713. Then, a cousin of him, Daniel Bernoulli (picture below) formally solved the problem, which was published in 1738. I cannot imagine how was a normal day of this amazing family, celebrating new discoveries, solutions, contributions... In total they were eight guys who contributed to the foundations of applied mathematics and physics. Amazing!
First, let's write the meaning of the word "paradox". Wikipedia says that Paradox is "a statement that apparently contradicts itself and yet might be true". This may sound confusing but maybe an example can help! What about the St. Petersburg paradox? Actually, this is also known as St. Petersburg lottery, and is as follows:
Imagine the typical game of tossing of coins. In this game, the coin is tossed until tails appears. Assume that our winnings increase, the longer the run of heads. So we win $1 if tails appears the first time, $2 if it come on the second, $4 on the third, and so on. The reward doubling each time. Also assume that you have to pay each time you play. The question is: what is a fair price for entry to the game?
The usual approach to solve it would be to calculate the game's expected reward. If that exceeds the entry price, then rationally you should agree to play, otherwise you shouldn't. This sounds reasonable, until we actually try to perform the calculation, when something very strange happens.
Each possible outcome has its own probability and reward: the probability that the game will end after one throw is 1/2, and this case, we would win $1. There is a probability of 1/4 that it will end after two throws, netting us $2. The probability is 1/8 after three throws, and we would get $4, and so on. Putting all these together, the expectation seems to come out as
(1/2 x $1) + (1/4 x $2) + (1/8 x $4) + (1/16 x $8) + ...
That is to say: $0.5 + $0.5 + $0.5 + $0.5 + ... In other words, the "expected" reward for this game is actually infinite! That's why this is called a "paradox". How can a simple game be infinitely profitable?Yet there is a sense in which it is correct. Again, it comes from repeated playing. Surprisingly, if you play the game often enough, then any entry price will indeed eventually represent good value. The catch is the number of rounds of the game needed. If the cost is $10 per game, you are likely to have to play over a million times before coming out in profit. If the cost is $100 per game, the lifetime of the universe is unlikely to be long enough. More about this and other paradoxes can be found in the book "Chaotic Fishponds and Mirror Universes" by Richard Elwes.
The Bernoulli family was involved in this problem. In fact, the problem was invented by Nicolas Bernoulli in 1713. Then, a cousin of him, Daniel Bernoulli (picture below) formally solved the problem, which was published in 1738. I cannot imagine how was a normal day of this amazing family, celebrating new discoveries, solutions, contributions... In total they were eight guys who contributed to the foundations of applied mathematics and physics. Amazing!
There are plenty of paradoxes in different fields of science. They give you the chance to imagine certain unexpected and hypothetical situations. Strange results can make you think twice, and that's good! This may have a certain link with a quote from Einstein:.
See you around!
Jesus
"If the facts don't fit the theory, change the facts"See you around!
Jesus